9Ji'!;i:'il|'!iU:|!iiii 


il!!:i;:ii!;!^l?iii.!:i 


c 


HYPERBOLIC  FUNCTIONS. 


MATHEMATICAL  MONOGRAPHS 

J.DITKD    BY 

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PUBLISHED  BY 
JOHN  WILEY  &  SONS,  Inc.,  NEW  YORK 

CHAP.MAN   &   HALL.  Limited.  LONDON 


MATHEMATICAL  MONOGRAPHS. 


EDITED    BY 


MANSFIELD   MERRIMAN  and  ROBERT   S.   WOODWARD. 


No.  4. 


HYPERBOLIC  FUNCTIONS. 


JAMES    McMAHON, 

LaIE  I'ROFEsb^K  OF  iU  A  1  H  EM  Al  ICb  IN   CoKNELL   UNIVERSITY. 


FOURTH    EDITION.   ENLARGED. 


NEW  YORK: 

JOHX    WILEY    &    SONS. 

London:    CHAPMAN  e^   HALL,    Limited. 


(X-cUl- 


Copyright,  1S96, 

BY 

MANSFIELD   MF.RRIMAN  and  ROBERT   S.   WOODWARD 

UNDF.K    THE    T    ]  LE 

HIGHER    MATHEMATICS. 

First  Edition,  September,  1896. 
Sjecond   Edition,  January,  1898. 
Third  Edition,   August,   1900. 
Fourth  Edition,  January,  1906. 


9/^7 


Printed  in  U.  S.  A. 


PRESS    OF 

BRAUNWOHTH    &    CO   .    INC 

eOOK    MANUFACTURERS 

BROOKLVN.    NEW  VCrtK 


EDITORS'   PREFACE. 


The  volume  called  Higher  Mathematics,  the  first  edition 
of  which  was  published  in  1896,  contained  eleven  chapters  by 
eleven  authors,  each  chapter  being  independent  of  the  others, 
but  all  supposing  the  reader  to  have  at  least  a  mathematical 
training  ecjuivalent  to  that  given  in  classical  and  engineering 
colleges.  The  pubHcation  of  that  volume  is  now^  discontinued 
and  the  chapters  are  issued  in  separate  form.  In  these  reissues 
it  will  generally  be  found  that  the  monographs  are  enlarged 
by  additional  articles  or  appendices  which  either  amplify  the 
former  presentation  or  record  recent  advances.  This  plan  of 
publication  has  been  arranged  in  order  to  meet  the  demand  of 
teachers  and  the  convenience  of  classes,  but  it  is  also  thought 
that  it  may  prove  advantageous  to  readers  in  special  lines  of 
mathematical  literature. 

It  is  the  intention  of  the  publishers  and  editors  to  add  other 
monographs  to  the  series  from  time  to  time,  if  the  call  for  the 
same  seems  to  warrant  it.  Among  the  topics  which  are  under 
consideration  are  those  of  eUiptic  functions,  the  theory  of  num- 
bers, the  group  theory,  the  calculus  of  variations,  and  non- 
Euchdean  geometry;  possibly  also  monographs  on  branches  of 
astronomy,  mechanics,  and  mathematical  physics  may  be  included. 
It  is  the  hope  of  the  editors  that  this  form  of  pubhcation  may 
tend  to  promote  mathematical  study  and  research  over  a  wider 
field  than  that  which  the  former  volume  has  occupied. 

December,  1905. 


742995 


AUTHOR'S   PREFACE. 


This  compendium  of  hyperbolic  trigonometry  was  first  published 
as  a  chapter  in  Merriman  and  Woodward's  Higher  Mathematics. 
There  is  reason  to  believe  that  it  supplies  a  need,  being  adaj)ted  to 
two  or  three  ditTerent  types  of  readers.  College  students  who  have 
had  elementary  courses  in  trigonometry,  analytic  geometry,  and  differ- 
ential and  integral  calculus,  and  who  wish  to  know  .something  of  the 
hyperbolic  trigonometry  on  account  of  its  important  and  historic  rela- 
tions to  each  of  those  branches,  will,  it  is  hoped,  find  these  relations 
presented  in  a  simple  and  comprehensive  way  in  the  first  half  of  the 
work.  Readers  who  have  some  interest  in  imaginaries  are  then  intro- 
duced to  the  more  general  trigonometry  of  the  complex  plane,  where 
the  circular  and  hyperbolic  functions  merge  into  one  class  of  transcend- 
ents, the  singly  periodic  functions,  having  either  a  real  or  a  pure  imag- 
inary period.  For  those  who  abso  wish  to  view  the  subject  in  some  of 
its  practical  relations,  numerous  applications  have  been  selected  so  as 
to  illustrate  the  various  parts  of  the  theory,  and  to  show  its  use  to  the 
physicist  and  engineer,  appropriate  numerical  tables  being  supplied  for 
these  purposes. 

With  all  these  things  in  mind,  much  thought  has  been  given  to  the 
mode  ot  approaching  the  subject,  and  to  the  presentation  of  funda- 
mental notions,  and  it  is  hoped  that  some  improvements  are  discerni- 
ble. For  instance,  it  has  been  customary  to  define  the  hyperbolic 
functions  in  relation  to  a  sector  of  the  rectangular  hyperbola,  and  to 
take  the  initial  radius  of  the  sector  coincident  with  the  principal  radius 
of  the  curve,  in  the  present  work,  these  and  similar  restrictions  are 
discarded  in  the  interest  of  analogy  and  generality,  with  a  gain  in  sym- 
metry and  simplicity,  and  the  functions  are  defined  as  certain  charac- 
teristic ratios  belonging  to  any  sector  of  any  hyperbola.  Such  defini- 
tions, in  connection  with  the  fruitful  notion  of  correspondence  of  points 
on  comes,  lead  to  simple  and  general  proofs  of  the  addition-theorems, 
from  which  easily  follow  the  conversion-formulas,  the  derivatives,  the 
Maclaurin  expansions,  and  the  ex{)onential  expressions.  The  proofs 
are  .so  arranged  as  to  apj)ly  equally  to  the  circular  functions,  regarded 
as  the  characteristic  ratios  belonging  to  any  elliptic  sector.  For  th(j.se, 
however,  who  mav  wish  to  start  with  the  exponential  expressions  as 
the  definitions  of  the  hyperl)olic  functions,  the  appropriate  order  of 
procedure  is  indicated  on  page  25.  and  a  direct  mode  of  l)ringing  such 
exponential  definitions  into  geometrical  relation  with  the  hvperbolic 
sector  is  shown  in  the  Appendix. 

December.  n)Oz,. 


CONTENTS„ 


Art.     I.  Correspondence  of  Points  on  Conics Page    ? 

2.  Areas  of  Corresponding  Triangles g 

3.  Areas  of  Corresponding  Sectors 9 

4.  Characteristic  Ratios  of  Sectorial  Measures    .      .           .      .  10 

5.  Ratios  Expressed  as  Triangle-measures 10 

6.  Functional  Relations  for  Ellipse 11 

7.  Functional  Relations  for  Hyperbola 11 

8.  Relations  between  Hyperbolic  Functions 12 

9.  Variations  of  the  Hyperbolic  P'unctions  .,..,.,  14 

10.  Anti  hyperbolic  Functions .      .  .16 

11.  Functions  of  Sums  and  Differences 16 

12.  Conversion  Formulas ,18 

13.  Limiting  Ratios    . .  19 

14.  Derivatives  of  Hyperbolic  Functions 20 

15.  Derivatives  of  Anti-hyperbolic  Functions 22 

16.  Expansion  of  Hyperbolic  Functions 23 

17.  Exponential  Expressions 24 

18.  Expansion  of  Anti-functions 25 

19.  Logarithmic  Expression  of  Anti-functions 27 

20.  The  Gudermanian  Function 28 

21.  Circular  Functions  of  Gudermanian 28 

22.  Gudermanian  Angle 29 

23.  Derivatives  of  Gudermanian  and  Inverse     ....  -30 

24.  Series  for  Gudermanian  and  its  Lnverse 31 

25.  Graphs  of  Hyperbolic  Functions ^2 

26.  Elementary  Integrals  ...            ...           3^ 

27.  Functions  of  Complex  Numbers             .     .           38 

28.  Addition  Theorems  for  Complexes       .,..,..  40 

29.  Functions  of  Pure  Imaginaries  .  .41 

30.  Functions  of  x+ty  in  the  Form  X  ^iV 43 

31.  The  Catenary' ....            .      .  47 

32    The  Catenary  of  Uniform  Strength    .      .           49 

33.  The  Elastic  Catenary 50 

34.  The  Tr.actory .      .  51 

35.  The  Loxodrome  .....,..,,,,            .  52 


6  CONTENTS. 

Art.  36    Combined  Flexure  and  Tension 53 

37.  Alternating  Currents 55 

38.  Miscellaneous  Applications 60 

39.  Explanation  of  Tables 62 

Table      I.  Hyperbolic  Functions 64 

II.  Values  of  cosh  {x^iy)  and  sinh  (x+iy) 06 

III.  Values  of  gdu  and  0'^ 70 

IV.  \'ALUES   of   gdw,   LOG  SINH  U,   LOG  COSH  U 70 

Appendix.   Historical  and  Bibliographical 71 

Exponential  Expressions  as  Definitions    ....  72 


Index 


73 


HYPERBOLIC    FUNCTIONS. 


Art.    1.      CORRESPONDENXE   OF    POINTS   ON   CONICS. 

To  prepare  the  way  for  a  general  treatment  of  the  hyper- 
bolic functions  a  preliminary  discussion  is  given  on  the  relations 
between  hyperbolic  sectors.  The  method  adopted  is  such  as 
to  apply  at  the  same  time  to  sectors  of  the  ellipse,  including 
the  circle;  and  the  analogy  of  the  hyperbolic  and  circular 
functions  will  be  obvious  at  every  step,  since  the  same  set  of 
equations  can  be  read  in  connection  with  either  the  h}'perbola 
or  the  ellipse.*  It  is  convenient  to  begin  with  the  theory  of 
correspondence  of  points  on  two  central  conies  of  like  species, 
i.e.  either  both  ellipses  or  both  hyperbolas. 

To  obtain  a  definition  of  corresponding  points,  let  (9,/4,, 
0J\  be  conjugate  radii  of  a  central  conic,  and  O^A^,  O^B^ 
conjugate  radii  of  any  other  central  conic  of  the  same  species; 
let /'j , /*,  be  two  points  on  the  curves;  and  let  their  coordi- 
nates referred  to  the  respective  pairs  of  conjugate  directions 
be  (^, ,  J',),  (.1', ,  J',);  tlien,  by  analytic  geometry, 

*The  hyperbolic  functions  are  not  so  named  on  account  of  any  analogy 
with  what  are  termed  Elliptic  Functions.  "  The  elliptic  integrals,  and  thence 
the  elliptic  functions,  derive  their  name  from  the  early  attempts  of  mathemati- 
cians at  the  rectification  of  the  ellipse.  ...  To  a  certain  extent  this  is  a 
disadvantage;  .  .  .  because  we  employ  the  name  hyperbolic  function  to  de- 
note cosh  M  sinh  «,  etc.,  by  analogy  with  which  the  elliptic  functions  would  be 
merely  the  circular  functions  cos  <p,  sin  (p,  etc.  .  ."  (Greenhill,  Elliptic 
Functions,  p.  175.) 


IlVPKKliOLIC    I-UNlTH'NS. 


(2) 


Now  if  the  .points  /'  ,  P^  be  so  situated  that 

the  cquahties  referring  to  sign  as  well  as  magnitude,  then  /•*, , 
P  are  called  corresponding  points  in  the  two  systems.  If  (^, , 
0^  be  another  pair  of  correspondents,  then  the  sector  and  tri- 

A, 


angle  P,0,Q,  are  said  to  correspond  respectively  with  the 
sector  and  t  iangle  /'„0.,0.^.  Tiiese  definitions  will  a[)pl}'  also 
when  the  conies  coincide,  the  points  P^  ,  P^  being  then  referred 
to  any  two  pairs  of  conjugate  diameters  of  the  same  conic. 

In  discussing  the  relations  between  corresp(~>nding  areas  it 
is  convenient  to  adopt  the  following  use  of  the  word  "  measure": 
The  measure  of  an\'  area  connected  with  a  given  central  conic 
is  the  ratio  which  it  bears  to  the  constant  area  of  the  triangle 
formed  by  two  conjngite  diameters  of  the  same  conic. 

i'^or  example,  the  measure  of  the  sector  yl^OJ\  is  the  ratio 
sector  A,0,P^ 
triangle  .r/?^, 


AREAS    OK    CORRESPONDING    SECTORS.  9 

and  is  to  be  regarded  as  positive  or  negative  according  as 
A^OJ^^  and  A^O^B^  are  at  the  same  or  opposite  sides  of  their 
common  initial  Hne. 

Art.  2.  Areas  of  Cokresponding  Triangles. 
The  areas  of  corresponding  triangles  have  equal  measures. 
For,  let  the  coordinates  of  P^,  Q^  be  (-f , ,  Ji),  (-t'/,  j/)'  ^^^^  ^^t 
those  of  their  correspondents/',,  Q^  be  (;f,,  jj,  (-^V- j/);  let  the 
tY\a.ng\cs  P^O.Q,,  I\O^Q^hc  Z, ,  T^,  and  let  the  measuring  tri- 
angles A^0^6\,  A^O^Bj  be  A',,  A',,  and  their  angles  &?, ,  a\] 
then,  by  analytic  geometr}%  taking  account  of  both  magnitude 
and  direction  of  angles,  areas,  and  lines, 

T,   ^  i{A\v!-  x/y,)  sin  a?,  ^  .r,  j/  _  .r/  y^^ 

7^  ^  i.rj'/-,r/j,)  sin  (k?^  _x^  j/  _  £^'_^^ 
A',  i^?/,  sin  &7,  rt,    <^,  tf,  b^ 

T         T 
Therefore,  by  (2),  -f  =  — \  (3) 

Art.  3.  Areas  of  Corresponding  Sectors. 
The  areas  of  corresponding  sectors  have  equal  measures. 
For  conceive  the  sectors  5,,  5,  divided  up  into  infinitesimal 
corresponding  sectors  :  then  the  respective  infinitesimal  corre- 
sponding triangles  have  equal  measures  (Art.  2);  but  the 
given  sectors  are  the  limits  of  the  sums  of  these  infinitesimal 
triangles,  hence 

5,        5, 

In  particular,  the  sectors  A^O.P^,  A^O^P^  have  equal  m.eas- 
ures ;  for  the  initial  points  A^,  A^  are  corresponding  points. 

It  may  be  proved  conversely  by  an  obvious  reductio  ad 
absurdum  that  if  the  initial  points  of  two  equal-measured 
sectors  correspond,  then  their  terminal  points  correspond. 

Thus  if  any  radii  0,A^,  O^A^  be  the  initial  lines  of  two 
equal-measured  sectors  whose    tei-minal   radii    are   O^P^,   O^P^^ 


10  HYPERBOLIC    FUNCTIONS. 

then  /",,  P^  are  corresponding  points  referred  respectively  to 
the  pairs  of  conjugate  directions  6^,^,,  (9,Z),,  and  O.^A^,  0^B^\ 
tliat  is, 

•^  _  ^2   y^  ^yj_ 

a^  ~  a,'     d,  ~  b: 

Prob.  I.  Prove  that  the  sector  P^O^Q^  is  bisected  by  the  Hne 
joining  O^  to  tlie  mid-point  of  P ^Q^^.  (Refer  the  points  P ^,  Q^,  re- 
spectively, to  the  median  as  common  axis  of  .v,  and  to  the  two 
opposite  conjugate  directions  as  axis  of  y,  and  show  that  P^,  Q^ 
are  then  corresjjonding  points.) 

Prob.  2.  Prove  that  the  measure  of  a  circuhTr  sector  is  equal  to 
the  radian  measure  of  its  angle. 

Prob.  3.  Find  the  measure  of  an  elliptic  quadrant,  and  of  the 
sector  included  by  conjugate  radii. 

Art.  4,    Char.acteristic  Ratios  of  Sectorial 
Measures. 

Let  A^O^P^  =  5,  be  any  sector  of  a  central  conic;  draw 
P^M,  ordinate  to  0,A^,  i.e.  parallel  to  the  tangent  at  A^; 
let  0,AI,  =  -r,,  J/,/^,  =J\,  O^A^  =  <?, ,  and  the  conjugate  radius 
O^B^=z^^•,  then  the  ratios  xja^.yjb^  are  called  the  charac- 
teristic ratios  of  the  given  sectorial  measure  SJK^.  These 
ratios  are  constant  both  in  magnitude  and  sign  for  all  sectors 
of  the  same  measure  and  species  wherever  these  may  be  situ- 
ated (Art.  3).  Hence  there  exists  a  functional  relation  be- 
tween the  sectorial  measure  and  each  of  its  characteristic 
ratios. 

Art.  5.     Ratios  Expressed  as  Triangle-measures. 

The  triangle  of  a  sector  and  its  complementary  triangle  are 
measured  by  the  two  characteristic  ratios.  For,  let  the  triangle 
A^O^P^  and  its  complementary  triangle  Pfi^B^  be  denoted   by 

r,,  r/;  then 

r,  _  hij,  sin  &7,  _  J ,    "^ 
K^  ~  \aj)^  sin  ct?,  ~  b^ 


t;  _\b^x^s\\\  &7, 


y  (5) 


K^       \ajh^  sin  &?,       a^     \ 


FUNCTIONAL    RELATIONS    FOR    ELLIPSE. 


11 


Art.  6.     FUxXCTIONal  Relations  for  Ellipse. 

The  functional  relations  that  exist  between  the  sectorial 
measure  and  each  of  its  ciiaracteristic  ratios  are  the  same 
for  all  elliptic,  in- 
cluding circular,  sec- 
tors (Art.  4).  Let/^, , 
P^  be  corresponding 
points  on  an  ellipse 
and  a  circle,  referred  o 
to  the  conjugate  di- 
rections O^A^,  O^B^,  and  O^  A^,0„B„,  tlie  latter  pair  being  at 
right  angles;  let  the  angle  A^O^P^  =■  0  in  radian  measure;   then 


-'  =  COS  -~ 

ci„  A  „ 


a;     hr: 

y,        .     S. 
iK  A, 


(6) 


[a,  =  d. 


hence,  in  the  ellipse,  by  Art.  3, 


—  =  cos  — - 
c^.  A , 


(7) 


Prob.  4.  Given  .v,  =  i<7,;  find  the  measure  of  the  elliptic  sector 
A^OiFu     Also  find  its  area  when  </,  =  4,  b^  r=  3,  &?  =  60". 

Prob.  5.   Find  the  characteristic  ratios  of  an  elii}  tic  sector  whose 
measure  is  jTT. 

Prol).  6.  Write  down  the  relation  between  an  elliptic  sector  and 
its  triangle.     (See  Art.  5.) 


Art.  7.    Functional  Relations  for  Hyperbola. 

The  functional  relations  between  a  sectorial  measure  and 
its  characteristic  ratios  in   the  case  of  the  hyperbola  ma}'  be 


written  in  the  form 

X 

a 


X  S       V  S 

-J  ~  cosh  — ^,    Y  =  sinh  — ;•, 
A.      ^,  AT. 


and  these  express  that  the  ratio  of  the  two  lines  on  the  left  is 
a  certain  definite  function  of  the  ratio  of  the  two  areas  on  the 
right.     These  functions  are  called  by  analogy  the  hyperbolic 


12  HYPERBOLIC    F UNCI  IONS. 

cosine  and  the  hyperbolic  sine.  Thus,  writing  u  for  SJK^,  the 
two  equations 

—  =  cosli  u,    V  =  smh  II  {%\ 

a,  b^  ^  ' 

serve  to  define  the  hyperbohc  cosine  and  sine  of  a  given  secto- 
rial measure  u\  and  the  h\-perbolic  tangent,  cotangent,  secant, 
and  cosecant  are  then  defined  as  follows: 


sinh  ?/  ,  cosh?^    ^ 


tanh  21  =  ^ , — .     coth  ti  -~ 


sech  71  = 


cosh  7('  smh  //' 

I  ,  I 


!-  (9) 


J 


The  names  of  these  functions  may  be  read  "  h-cosine," 
or  "hyper-cosine,"  etc.  (See  "  angloid  "  or  "hyperbolic 
angle,"  p.  ys-) 

Art.  8.  Relations  among  Hyperbolic  Functions. 
Among  the  six  functions  there  are  five  independent  rela- 
tions, so  that  wlien  the  numerical  value  of  one  of  the  functions 
is  given,  the  values  of  the  other  five  can  be  found.  Four  of 
these  relations  consist  of  the  four  defining  equations  (9).  The 
fifth  is  derived  from  the  equation  of  the  h}-perbola 

X '       vr 

_• -Ll  =  I 

a,        b, 
giving 

cosh'//  —  sinh";/  =  i.  (lo) 

By  a  combination  of  some  of  these  equations  other  subsidi- 
arv  relations  may  be  obtained;  thus,  dividing  (10)  successively 
by  coslr  //,  sinh'  //,  and  applying  (9),  give 

I  —  tanh'  u  =  sech"  //,  ) 

(II) 

coth'  //  —  I  =  csch'  //.  ) 

Equations  (9),  (lo),  (ii)  will  readily  serve  to  express  the 
value  of  any  function  in  terms  of  an}'  other.  For  example, 
when  tanh  //  is  given, 

coth  u  —  ,  sech  //  =  ^/ \  —  tanh'//, 

tanh  u 


RELATIONS    BETWEEN    HYPERBOLIC    FUNCTIONS. 


13 


cosh  n  =. 


csch  // 


V   I  —  Vax\\\^u 

yf  I   —  tanh'z/ 
tanh  11 


sinh  u 


taiih  u 


V 


tanh'^ 


The  ambiguity  in  the  sign  of  tlie  square  root  may  usually 
be  removed  by  the  following  considerations  :  The  functions 
cosh  71,  sech  u  are  always  positive,  because  the  primary  char- 
acteristic ratio  x,/(?,  is  positive,  since  the  initial  line  O^A^  and 
the  abscissa  O^M^  are  similarly  directed  from  O^,  on  which- 
ever branch  of  the  hyperbola  P^  may  be  situated;  but  the  func- 
tions sinh  //,  tanh  u,  coth  //,  csch  //,  involve  the  other  charac- 
teristic ratio  vjl\,  which  is  positive  or  negative  according  as 
}\  and  l\  have  the  same  or  opposite  signs,  i.e.,  as  the  measure 
//  is  positive  or  negative;  hence  these  four  functions  are  either 
all  positive  or  all  negative.  Thus  when  any  one  of  the  func- 
tions sinh//,  tan.h //,  csch//,  coth//,  is  given  in  magnitude  and 
sign,  there  is  no  ambiguity  in  the  value  of  any  of  the  six 
h\-perbolic  functions ;  but  when  either  cosh  //  or  sech  //  is 
given,  there  is  ambiguity  as  to  whether  the  other  four  functions 
shall  be  all  positive  or  all  negative. 

The  hyperbolic  tangent  may  be  expressed  as  the  ratio  of 
two  lines.     For  draw  the  tangent 
line  AC ^=^  t  \  then 


y     X      ay 

tanh  u  =z-L  :_  =  -.- 

h      a       0    X 


(I2)   o 


The  hyperbolic  tangent  is  the  measure  of  the  triangle  OAC. 


For 


OAC      at       f 

OAB  =  a^=J  =  ''''^'''- 


(13) 


Thus  the  sector  AOP,  and  the  triangles  AOP,  POP,  AOC, 
are  proportional  to  //,  sinh  ?(,  cosh  //,  tanh  u  (eqs.  5,  13)  ;  hence 

sinh /^  >  «>  tanh//.  (14) 


14 


HYPERBOLIC    FUNCTIONS. 


Prol).  7.  Express  all  the  hyperbolic  functions  in  terms  ot  sinh  u. 
Given  cosh  u  =  2,  find  the  values  of  the  other  functions. 

Prob.  8.  Prove  from  ecjs.  10,  11,  that  coslw/>  sinh//,  cosh//'>i, 
tanh  //  <  I,  sech  //  <   i. 

Prob.  9.  In  the  figure  of  Art.  i,  let  OA  —  2,  OB=i,  AOB  —  60", 
and  area  of  sector  AOP  =  3;  find  the  sectorial  measure,  and  the 
two  characteristic  ratios,  in  the  elliptic  sector,  and  also  in  the  hyper- 
bolic sector;  and  find  the  area  of  the  triangle  AOP.  (Use  tables  of 
cos,  sin,  cosh,  sinh.) 

Prob.  10.  Show  that  coth  ii,  sech  le,  csch  u  may  each  be  ex- 
pressed as  the  ratio  of  two  lines,  as  follows:  Let  the  tangent  at  P 
make  on  the  conjugate  axes  OA,  OB,  intercepts  OS  =  w,  OT  —  n\ 
let  the  tangent  at  B,  to  the  conjugate  hyperbola,  meet  OP  in  R^ 
making  BR  =  /;  then 

coth  //  =  //<?,     sech  ii.  =  m/a,     csch  //  =  n/b. 

Prol).  II.  The  measure  of  segment  AMP  is  sinh  11  cosh  u  —  11. 
Modify  this  for  the  ellipse.  Modify  also  eqs.  10-14,  ^'^d  probs. 
8,  10. 

Art,  9.    Variations  of  the  Hyperbolic  Funxtioxs. 

Since  the  values  of  the  hyperbolic  functions  depeiul  only 
on  the  sectorial  measure,  it  is  convenient,  in  tracing  their  vari- 
ations, to  consider  only  sectors  of  one 
half  of  a  rectangular  iiyperbola,  whose 
conjugate  radii  are  equal,  and  to  take  the 
principal  axis  OA  as  the  common  initial 
line  of  all  the  sectors.  The  sectorial 
measure  ;/  assumes  every  value  from  —  00, 
through  o,  to  -|-  00  ,  as  the  terminal  point 
P  comes  in  from  infinit)'  on  the  lower 
branch,  and  passes  to  infinity  on  the  upper 
branch  ;  that  is,  as  the  terminal  line  OP 
swings  from  the  lower  asymptotic  posi- 
tion y  =  —  X,  to  the  upper  one,  y  =  x.  It  is  here  assumed, 
but  is  proved  in  Art.  17,  that  the  sector  AOP  becomes  infinite 
as  /-"passes  to  infinity. 

Since  the  functions  cosh  7/,  sinh  u,  tanh  ?/,  for  any  position 


Variations  of  the  hyperbolic  functions.  16 

of  OP,  are  equal  to  the  ratios  of  x,  y,  t,  to  the  principal  radius 
a,  it  is  evident  from  tlie  figure  that 

cosh  0=1,     sinh  0  =  0,     tauh  0=0,  (15) 

and  that  as  u  increases  towards  positive  infinity,  cosh  //,  sinh  u 
are  positive  and  become  infinite,  but  tanh  11  approaches  unity 
as  a  limit ;  thus 

cosh  CO   =  00  ,     sinh  00   =  co  ,     tanh  00   =  i.  (16) 

Again,  as  n  changes  from  zero  towards  the  negative  side, 
cosh  H  is  positive  and  increases  from  unity  to  infinit}',  but 
sinh  u  is  negative  and  increases  numerically  from  zero  to  a 
negative  infinite,  and  tanh  u  is  also  negative  and  increases 
numerically  from  zero  to  negative  unity  ;  hence 

cosh  (— 00  )  =  CO  ,    sinh  (— CO  )  =  —  CO  ,    tanh  (— 00  )  =  —  i.  (17) 

For  intermediate  values  of  //  the  numerical  values  of  these 
functions  can  be  found  from  the  formulas  of  Arts.  16,  17,  and 
are  tabulated  at  the  end  of  this  chapter.  A  general  idea  of 
their  manner  of  variation  can  be  obtained  from  the  curves  in 
Art.  25,  in  which  the  sectorial  measure  u  is  represented  by  the 
abscissa,  and  the  values  of  the  functions  cosh  //,  sinh  ii,  etc., 
are  represented  by  the  ordinate. 

The  relations  between  the  functions  of  —  11  and  of  ii  are 
evident  from  the  definitions,  as  indicated  above,  and  in  Art.  8. 
Thus 

cosh  (—  u)  =  -["  cosh  u,     sinh  ( —  ?0  —  ~  sinh  n,   \ 

sech  {—  n)  =  -j-  sech  u,     csch  [—  i/)  =:  —  csch  ?/,    >-     (18) 

tanh  (—  ii)  =  —  tanh  u,     coth  (—  //)  =  —  coth  //.   j 

Prob.  12.  Trace  the  changes  in  sech  //,  coth  7/,  csch  //,  as  1/  passes 
from  —  CO  to  +  00 .  Show  that  sinh  u,  cosh  //  are  infinites  of  the 
same  order  when  u  is  infinite.  (It  will  appear  in  Art.  17  that  sinh 
u,  cosh  t/  are  infinites  of  an  order  infinitely  higher  than  the  order 
ofu.) 

Prob.  13.  Applying  eq.  (12)  to  figure,  page  14,  prove  tanh  i/,  = 
tan  A  OF. 


16  Hyperbolic  functions. 

Art.  10.    Anti-hyperbolic  Functions. 
X  y       ■  ^ 

Tlie  equations  -  =  cosh  u,    -j  =  sinh  //,   7  =  tanli  u,  etc., 
*  a  a  b 

may  also  be   expressed  by  the    inverse  notation  ?^  =  cosh"^  — , 

_  y  t 

u  =  sinh  ^-7,  u  ^=  tanh  '— ,  etc.,  which    may  be    read:    "  ;^    is 

the  sectorial  measure  whose  hyperbolic  cosine  is  the  ratio  x  to 
«,"  etc. ;  or  "  u  is  the  anti-h-cosine  of  x/a''  etc. 

Since  there  are  two  values  of  7i,  with  opposite  signs,  that 
correspond  to  a  given  value  of  cosh  u,  it  follows  that  if  u  be 
determined  from  the  equation  cosh  ti  =  m,  where  m  is  a  given 
number  greater  than  unity,  u  is  a  two-valued  function  of  w. 
The  symbol  cosh  '  m  will  be  used  to  denote  the  positive  value 
of  //  that  satisfies  the  equation  cosh  u  —  vi.  Similarly  the 
symbol  sech"*  vi  will  stand  for  the  positive  value  of  11  that 
satisfies  the  equation  sech  21  =  ;//.  The  signs  of  the  other 
functions  sinii"'w,  tanh"';;/,  coth~' ;;/,  csch"' ;;/,  are  the  same 
as  the  sign  of  ;;/.  Hence  all  of  the  anti-hyperbolic  functions 
of  real  numbers  are  one-valued. 

Prob.  14.  Prove  the  following  relations: 

cosh"';//  =  sinh"'  V m^  —  i,     sinh"';;/  =   ±  cosh''  V;;/"  -j-  i, 
'he.  upper  or  lower  sign  being  used  according  as  ;;/  is  positive  or 
negative.     Modify  these  relations  for  sin "' ,  cos"' . 

Prob.  15.  In  figure,  Art.  i,let  OA  =  2,0B  =  i,AOB  =  60°;  find 
the  area  of  the  hyperbolic  sector  A  OP,  and  of  the  segment  AMP, 
if  the  abscissa  of  P  is  3.     (Find  cosh"'  from  the  tables  for  cosh.) 

Art.  11.    Functions  of  Sums  and  Differences. 

(a)  To  prove  the  difference-formulas 

sinh  (//  —  7')  =  sinh  //  cosh  t>  —  cosh  //  sinh  7>, ) 

(  (19) 

cosh  (7/  —  7')  =  cosh  ;/  cosh  7'  —  sinh  //  sinh  1'.  ) 

Let  OA  be  any  radius  of  a  hyperbola,  and  let  the  sectors  AOP, 
AOQ  have  the  measures  //,  v\  then  //  —  v  is  the  measure  of  the 
sector  QOP.  Let  OB,  OQ'  be  the  radii  conjugate  to  OA,  OQ; 
and  let  the  coordinates  of  P,  Q,  Q'  be  (^,  ,J,),  i-^,  y),  (^'>  j') 
with  reference  to  the  axes  OA,  OB;  then 


FUNCTIONS  OF  SUMS  AND  DIFFERENCES. 


17 


.   ,    ,           ,         .   ,    sector  (9(9/^      trianHe  (9(9/'    ..    ^ 
Sinn  {ti  —  V)  =  siiih  -^ — ■  = ^    —     [Art.  5. 

ii^Ji—  -i'jO  sin  00 j\  X       y  x^ 

^aj)^  sin  00  b^  a^      b^  a^ 

=  sinh  u  cosh  v  —  cosh  ii  sinh  v\ 


cosh  (?^  —  ■z')  =  cosh 


sector  OOP      trianorle  POO' 


K  K 

K't'y-^.'O  sin  03 _  y'  ,r,        y^x' 


[Art.  5. 


but 


2,aJ^^  sin  gj? 

y' 

X 

—     > 

X'  _y 
a.       b! 

b.  a. 


b.  a, ' 


(20) 


since  Q,  Q'  are  extremities  of  conjugate  radii ;  hence 

cosli  {11  —  7')  =  cosh  n  cosh  v  —  sinh  u  sinh  v. 

In  the  figures  11  is  positive  and  v  is  positive  or  negative. 
Other  figures  may  be  drawn  with  u  negative,  and  the  language 
in  the  text  will  apply  to  all.  In  the  case  of  elliptic  sectors, 
similar  figures  may  be  drawn,  and  the  same  language  will  apply, 
except  that  the  second  equation  of  (20)  will  be  x' /a^  =  — //^,; 
therefore 

sin  (;/  —  v)  =  sin  ?/  cos  2'  —  cos  ?/  sin  t>, 

cos  {?(  —  I')  =  cos  7/  cos  V  -\-  sin  u  sin  v. 

(b)  To  prove  the  sum-formulas 

sinh  (7/  --\-  v)  =:  sinh  u  cosh  v  -{-  cosh  7/  sinh  ?', 
cosh  (//  -j-  v)  =  cosh  u  cosh  t  -{-  sinh  ?/  sinh  2'. 

These  equations  follow  from  (19)  by  changing  v  into  —  v, 


(21) 


18 


HYPERBOLIC    FUNCTIONS. 


and  then  for  sinh  (— f),  cosh  (— z'),  writing  —  sinh  t^,   cosh  z> 
(Art.  9,  eqs.  (i8)j. 


(c)  To  prove  that  tanh  {u  ±_  v)  = 


tanh  21  ±  tanh  v 
I  ±tanh  71  tanh  v 


(22) 


Writing  tanh  (u  ±  v)  =  ■ -— ^,   expanding  and  dividing 

cosh  {n  ±  T'j         ^  ^  ^ 

numerator  and  denominator  by  cosh  ?/  cosh  v,  eq.  (22)  is  ob- 
tained. 

^  Prob.  16.  Given  cosh  ?/  —  2,  cosh  v  =  3,  find  cosh  (//  +  v). 

Prob.  17.  Prove  the  following  identities: 
^    I.   sinh  2//  =  2  sinh  //  cosli  ?/. 

1^    2.    cosh  2u  =  cosh'/c  +  sinh'/^  =  i  -|-  2  sinh''  //  =  2  cosh^  u  —  i. 
'■^  3.   I  +  cosh  //  =  2  cosh'  4«,     cosh   /^  —  i  =  2  sinh^  -^u. 

sinh  //       _  cosh  ?'  —  i  _  /cosh  ?/  —  i\* 
I  +  cosh  //  sinh  u  \cosh  //  -\-  1/ 

.   ,  2  tanh  ?/  ,  T  -*-  tanh^  « 

5.  sinh  2U   = 


4.  tanh  ^u  = 


cosh    2tt 


I  —  tanh'' «'  '"        1   —  tunh'  u' 

6.  sinh  3/('   =  3  sinh  /^  +  4  sinh"  u,  cosh  3//  =  4  cosh'«  —3  cosh  a. 

,      .   ,  I  +  tanh  ^u 

7.  cosh  u  4-  sinh  ?/  = ,    :  -. 

'  I  —  tanh  ^u 

8.  (cosh  u  -f-  sinh  //)(cosh  Z'  +  sinh  Z')  =  cosh  {u  -\-  ?')  -f-  sinh  {u  -f  7')- 

9.  Generalize  (8);  and  show  also  vvhatrit  becomes  when  u  =  v^  .  .  , 

10.  sinh^v  cosj'  +  cosh^v  sin^  =  sinhV  -\-  sin^'j'. 

11.  cosh"'w  ±  cosli"';/  =  cosh~'Lw«  ±   V  (w'  —  i)(«'— i)j. 

12.  sinh"' w  ±  sinh"'//  =  sinh''|  w  y  i  -f-  «'  ±  ;/ y  i  +  m'j. 

Prob.  18.   What  modifications  of  signs  are  required  in  (21),  (22), 
in  order  to  pass  to  circular  functions  ? 

Prob.  19.   Modify  the  identities  of  Prob.  17  for  the  same  purpose. 


Art.  12.    Conversion  Formulas. 
To  prove  that 

cosh  7/,-|-  cosh  ?{,  —  2  cosh  ^(//.-f"  ''j)  cosh  K//,—  ?/,)» 
cosh  71,—  cosh  7/ J  =  2  sinh  f(//,  -j-^/Jsinh  i{7/,—  ?/,), 
sinh  71,  -\-  sinh  //,  =  2  sinh  ^(//,  -f  '0  cosh  ^u,—  ?/,),   j 
shih  «.  —  sinh  //,  =  2  cosh  ^(//,  -["  '^)  ■'^'"'1  aC'^  —  ''''o)-  J 


(23) 


LIMITING    RATIOS.  19 

From  the  addition  formulas  it  follows  that 

cosh  {u  -\-  v)  -j-  cosh  (//  —  v)  =  2  cosh  ji  cosh  v, 

cosh  [h  -\-  v)  —  cosh  {u  —  f)  =  2  sinh  u  sinh  v, 

siiih  [h  -{-  v)  ~\-  sinh  {u  —  7')  =  2  sinh  u  cosh  v^ 

■  sinh   (//  -{-v)  —  sinh  {u  —  z/)  =  2  cosh  Ji  sinh  ?', 

and  then  by  writing  u  -\-  v  =  //,  ,    u  —  v  zz^  n^ ,    u  =  ^(;/,  -f~  ^^)> 
^1  =  ^(;/,  —  z/^),     these  equations  take  the  form  required. 

Prob.  20.  In  passing  to  circular  functions,  show  that  the  only 
modification  to  be  made  in  the  conversion  formulas  is  in  tlie  alge- 
braic sign  of  the  right-hand  member  of  the  second  formula. 

_.     ,  ^.       ...    cosh  2U  +  cosh  AV  cosh  2U  -\-  cosh  4?^ 

Prob.    21.     Simplify     -r— ; ; r-; ,  , : ■• 

sinh  2U  -\-  smh  47;  cosh  2«  —  cosh  4^ 

Prob.  22.   Prove  sinh^x  —  sinh^'j^  =  sinh  (.v  -\-y)  sinh  {x  — y). 
Prob.  23.   Simplify  cosh^v  cosh^j'  ±  sinhlv  sinh'j'. 
Prob.  24.  Simplify  cosh^a*  cos^>'  -f-  sinh^x  sin'_y. 

Art.  13.     Limiting  Ratios. 
To  find  the  limit,  as  u  approaches  zero,  of 
sinh  u  tanh  ii 

U  II 

which  are  then  indeterminate  in  form. 

By  eq.  (14),  sinh  ii^  u~>  tanh  ti ;  and  if  sinh  ;/  and  tanh  Ji 
be  successively  divided  by  each  term  of  these  inequalities,  it 
follows  that 

sinh  II  . 

I   < <  cosh  u, 

u 

.  tanh  u 

sech  II  <  <  L* 

ti 

but  when  u-^O,  cosh  u  ^  i,  sech  ti  ^  i,  hence 

lim.    sinlw/^^^  ][,Ti.    tanh  u    ^  ^  .. 

u  =  o     u  '        u  ^o       II 


20 


HYPERBOLIC    FUNCTIONS. 


Art.  14.    Derivatives  of  Hyperbolic  Functions. 
To  prove  that 

^(sinh  u) 


{d) 


du 

^^(cosh  //) 
du 

</(tanh  n) 
du 

d{sQz\\  ii) 
du 

d{QO\.\\  II) 

du 

^/(csch  7/) 
du 


=   cosh  u, 

=  sinh  «, 

=  secli^  u, 

=  —  sech  7/  taiih  u, 

=■  —  csch' ;/, 

=  —  csch  u  coth  u. 


{a)  Let  y  =  sinh  7i, 

Ay  =  sinh  (u  -{-  Au)  —  sinli  7i 

z=  2  cosh  ^{2u  -\-  J7()  sinh  ^^u, 

Ay             1/11.  N^i'ih  ^z/« 
-f^  =  cosh  in  -f  ^  J//)  — - . . 

A7l  V        I      -         /        ij^ 

Take  the  limit  of  both  sides,  as  An  =  o,  and  put 

Ay       dy       ^'(sinh  7i) 

lim.  -7-  =  "T  = 'J ' 

A  u       d7i  d7t 

lim.  cosh  (//  -\-  \A7i)  =  cosh  u, 

sinli  iz/;/ 

=  I  ;         (see  Art.  13) 

=:  cosh  u. 


lim 
then 


(-/(sinh  77) 


du 


ip)  Simihir  to  {a). 

c/(tanh  7<)        d     sinh  ;/ 


{c) 


du  d7i '  cosh  u 

cosh"  u  —  sinh'  u 
cosh"  71 


(25) 


cosli"  u 


=  sech'  71. 


DERIVATIVES   OF    HYPERBOLIC    FUNCTIONS.  21 

{d)     Similar  to  (c). 

^/(sech  u)        d  I  sinh  n 

\e)      — — ; =  -J- . -. —  =  — : .:—  =:  —  scch  u  tanh  u. 

ail  an     cosh  u  cosh  u 

{/)     Similar  to  (r). 

It  thus  appears  that  the  functions  sinh  ;/,  cosh  u  reproduce 
themselves  in  two  differentiations  ;  and,  similarly,  that  the 
circular  functions  sin//,  cos//  produce  their  opposites  in  two 
differentiations.  In  this  connection  it  may  be  noted  that  the 
frequent  appearance  of  the  hyperbolic  (and  circular)  functions 
in  the  solution  of  physical  problems  is  chiefly  due  to  the  fact 
that  they  answer  the  question :  What  function  has  its  second 
derivative  equal  to  a  positive  (or  negative)  constant  multiple 
of  the  function  itself?  (See  Probs.  28-30.)  An  answer  such  as 
y  =  cosh  inx  is  not,  however,  to  be  understood  as  asserting  that 
inx  is  an  actual  sectorial  measure  and  j  its  characteristic  ratio; 
but  only  that  the  relation  between  the  numbers  nix  and  y  is  the 
same  as  the  known  relation  between  the  measure  of  a  hyper- 
bolic sector  and  its  characteristic  ratio;  and  that  the  numerical 
value  oi  y  could  be  found  from  a  table  of  hyperbolic  cosines. 

Prob.  25.  Show  that  for  circular  functions  the  only  modifica- 
tions required  are  in  the  algebraic  signs  of  (/'),  {d). 

Prob.  26.  Show  from  their  derivatives  which  of  the  hyperbolic 
and  circular  functions  diminish  as  //  increases. 

Prob.  27.  Find  the  derivative  of  tanh  //  independently  of  the 
derivatives  of  sinh  //,  cosh  //. 

Prob.  28.  Eliminate  the  constants  by  differentiation  from  the 
equation  y  ^=  A  cosh  mx  +  B  sinh  ?nx,  and  prove  that  d'^y/dx^  =  ni'y. 

Prob.  29.  Eliminate  tlae  constants  from  the  equation 
y  ^  A  cos  mx  +  B  sin  mx, 

and  prove  tliat  d'^y/dx'  =  —  my. 

Prob.  30.  Write  down  the  most  general  solutions  of  the  differen- 
tial equations 

^y  ,         dy  ^         d'y 

d?="'^^  d?  =  -"'^'^  dx-^  =  "'y- 


23  HYPERBOLIC    FUNC'J  IONS. 

Art.  15.    Derivatives  of  Anti-iivi-erbolic  Functions. 
^/(sinh"'  x)  I 


(0 

(/) 


c/x  vV  +  l' 

c/{cosh~'  x)  _  I 

dx  ~^/P"^' 

^(tanh~'  ;i-)  I 

I      —    X'j.x<i 

L_1 

I 


^^ 

./(coth  - 

x) 

./^ 

c/(sech   ' 

-r) 

^.jr 

rt'(csch- 

^) 

^  i  I  -  x' 
I 


^/^ 


(26) 


X  V  x'  -\-  I 

(<?)  Let     71  =  sinh"'  ,i-,     then  x  —-  sinli  ;^  dx  ~  cosh  //  (T';/ 


=  |/i  +  sinh'  7/  rt'/^  =  Vi  +  A-^  ^//^     rt';/  =  dx/  I  i  ^  ;r\ 
(<^)  Similar  to  {a), 
{c)  Let     7/  =  tanh"'  ,r,     then  x  =  tanh  /^  rt'x  =  sech""  //  du 

=  (i  —  tanh"  7{)dH  =  (i  —  x")du,     du  =  dx/i  —  x''. 
(d)  Similar  to  (c). 

^  '  dx  dx  \  xi       x    I    \x         I 

(/)  Similar  to  (r). 


X  \  I—  x' 


Prob.   31.   Prove 
^(sin~'  x)  _ 


1/1  -  .;c^' 


fl'(cos-'a-)  _  _  I 

dx        ~        4/1  -  x" 


a'(tan~'  •^)  _       i  ^'(cot"'  x)  _  i 

^"jf  I   -j-  x"  dx  14--^' 


EXPANSION    OF    HYPERBOLIC    FUNCTIONS. 


23 


Prob.  32.  Prove 

a  Sinn     —  = 
a 


^tanh"'  - 


aJx 


^cosh"'-  = 


dx 


<^         \/x'  -  a" 

,       ,    ,  -^v  a^x 

^7  coth"  —  = ^ 2 

a  X   —  a 


Prob.  2)Z'  Find  ^/(sech~'  x)  independently  of  cosh   '  x. 

Prob.  34.  When  tanh~'  x  is  real,  prove  that  coth~^  x  is  ima^^ 
nary,  and  conversely;  except  when  x  =  i. 

^     ,  T-     ,  sinh-'jc       cosh"' jr 

Prob.   -^S-  Evaluate      — ; ,     — ; ,     when  a:  =  00. 

"'^  log  X    '        log ;»: 


Art.  16.    Expansion  of  Hyperbolic  Functions. 


For  this  purpose  take  Maclaurin's  Theorem, 

/{u)  =  /(o)  +  ^//'(o)  +  ^\  /r/'(o)  +  ij  uy'io)  +  .  .  ., 

and  put  /{ii)  =  sinh  ?/,     f\ii)  =  cosli  u,    f''{u)  =  sinh  «, 
then         f{p)  =  sinh  0  =  0,    /'(o)  =  cosh  o  =  i,  .  .  .; 


hence  sinh  71 

and  similarly,  or  by  differentiation. 


"+?"■  + 51"'+ 


cosh  7/  =  I  -|- 


u^  +  . 


(27) 


(28) 


By  means  of  these  series  the  numerical  values  of  sinh  u, 
cosh  7/,  can  be  computed  and  tabulated  for  successive  values  of 
the  independent  variable  7i.  They  are  convergent  for  all  values 
of  21,  because  the  ratio  of  the  nth  term  to  the  preceding  is  in 
the  first  case  u"" /{2n  —  \){2n  —  2),  and  in  the  second  case 
«y(2«  —  2)(2«  —  3),  both  of  which  ratios  can  be  made  less  than 
unity  by  taking  n  large  enough,  no  matter  what  value  u  has. 
Lagrange's  remainder  shows  equivalence  of  function  and  series. 


24  HYPERBOLIC    FUNCTIONS. 

From  these  series  the  following  can  be  obtained  by  division  : 
tanh  u  =  u-  \u'  +  ^V'^  -  31^'^'  +  •  •  •    ~ 


sech  u  =  \  —  ^ii"  -)-  ^u*  —  yVo?/'  +  .  . 

U  COth  «  =  I   +  \H^  -  ^\U'  +  ^f  5/^'-  .  .   . 

u  csch  u  =1-  ^i^^^l^u'-^-^\'^u'-{-.. 


(29) 


These  four  developments  are  seldom  used,  as  there  is  no 
observable  law  in  the  coefficients,  and  as  the  functions  tanh  u, 
sech  u,  COth  u,  csch  ?/,  can  be  found  directly  from  the  previously 
computed  values  of  cosh  u,  sinh  ti. 

Prob.  36.  Show  that  these  six  developments  can  be  adapted  to 
the  circular  functions  by  changing  the  alternate  signs. 


Art.  17.     Exponential  Expressions. 


Adding  and  subtracting  (27),  (28)  give  the  identities 


cosh  u  -\-  sinh  u  ■=  \  -\-  u  -\-  — «'  -| -?/'  +  -7?^^  +  .  .  .  =  ^'', 

2!  '\ .  4! 


cosh  u  —  sinh  ti  —  \  —  u  A^ -u^  —  —7/'  -|-  — u^  —  .  .  .  =  e~", 


r>\ 


3 


4! 


hence    cosh  u  =  ^i^"  -\-  e  "),     sinh  u  =  ^(r"  —  e'"), 

e"  —  €"  .  2  r    (30) 


tanh  u  = 


sech  2i  = 


-,     etc. 


The  analogous  exponential  expressions  for  sin  ?/,  cos  u  are 


cos 


u  =  \e"'  -i-e^"'),     sin  u  =  —{e"'  —  ^-'"■),       {i  =  V  —  i) 


where  the  symbol  r"'  stands  for  the  result  of  substituting  7^z  for 
X  in  the  exponential  development 

This  will  be  more  fully  explained  in  treating  of  complex 
numbers,  Arts.  28,  2p. 


EXPANSION    OF    ANTI-FUNCTIONS.  tb 

Prob.  37.  Show  that  the  properties  of  the  liyperbolic  functions 
could  be  placed  on  a  purely  algebraic  basis  by  starting  with  equa- 
tions (30)  as  their  definitions  ;  for  example,  verify  the  identities  : 

sinh  (—//)  =  —  sinh  n,     cosh  (—//)  =  cosh  //, 

cosh^  u  —  sinh^  //=  1,     sinh  {u  -(-<')  =  sinh  //  cosh  v  -f  cosh  u  sinh  z', 

^/^(cosh  mil)  ^/'"(sinh  mu)  ,    .   , 

r-j =  m  cosh  mil,     -— =  m^  sinh  mu. 

du  dll 

Prob.  38.   Prove  (cosh  11  -\-  sinh  11)"  =  cosh  nu  -)-  sinh  7iu. 

Prob.  39.  Assuming  from  Art.  14  that  cosh  «,  sinh  u  satisfy  the 
differential  equation  //V/^/«'^  =i',  whose  general  solution  may  be 
written  y  —  ^e"  +  Be'",  where  yl,  B  are  arbitrary  constants  ;  show 
how  to  determine  A,  B  in  order  to  derive  the  expressions  for  cosh  //, 
sinh  //,  respectively.     [Use  eq.  (15).] 

Prob.  40.  Show  how  to  construct  a  table  of  exponential  func- 
tions from  a  table  of  hyperbolic  sines  and  cosines,  and  vice  versa. 

Prob.  41.  Prove  u  =  log^  (cosh  u  -\-  sinh  //). 

Prob.  42.  Sliow  that  the  area  of  any  hyperbolic  sector  is  infinite 
when  its  terminal  line  is  one  of  the  asymptotes. 

Prob.  43.   From  the  relation  2  cosh  u  —  e"  -f-  e'"  prove 

2''~'(cosh //)"  =  cosh  ////  +  ;/ cosh  {11  —  2)11 +  hi{/i—i)  cosh  (//— 4)«  +  ..., 

and  examine  the  last  term  when  «  is  odd  or  even. 

Find  also  the  corresponding  expression  for  2""'  (sinh  11)". 

Art.  18.     Expansion  of  Anfi-Functions. 


<'/(sinh  '  -t')  _  I 


Since       -'^ ~, — - — -  =  — =  (i  4-  x)-^ 

ax  ./-    .      .2  /     • 


12,134       1356- 
=  I x^  A--  ~  x' -i  I  x'  -f- 

2  24  246 


hence,  by  integration, 


23^245        2467^  ^^  ^ 

the   integration-constant   being    zero,    since    sinh  '  x  vanishes 
with  X.     This  series  is  convergent,  and  can  be  used  in  compu- 


26 


HYPERBOLIC    FUNCTIONS. 


tation,  only  when   x  <  i.      Another   series,  convergent   when 
X  >  I,  is  obtained  by  writing  the  above  derivative  in  the  form 


4sinh-'  x)       ,  5   ,      ,    .        i(      ,     i\'"* 


1 1+-^ 1 _L3 5 
2  x"       24  x'       246a- 


'•+•■•]• 


.,1  ^11  |II  I3I1I35I  /x 

.-.  sinh-' ^=  C+log^H ^ —  ^_-^^_-2.1  .         (32) 

'  2  2.r'     2  4  4,v*     2  4  6  6x'  ' 


where  C  is  the  integration-constant,   which  will  be  shown   in 
Art.  19  to  be  equal  to  log,  2. 

A  development  of  similar  form  is  obtained  for  cosh~'.r;  for 


^(cosh-'  x)       ,  ,       .    ,       I  /  I  \-* 


dx 


X\.     ^  2   X'^  2   AX'^  2   Afi  x'^'        S 


hence 


I    I 


4 
I  3    I 


4 
T  3  5    I 


cosh-;.'=r+log^'--^,--^-,--^^-.-...,    (33) 

in  which  C  is  again  equal  to  log,  2  [Art.  19,  Prob.  46].  In 
order  that  the  function  cosh"'.i'  maybe  real,  ;ir  must  not  be 
less  than  unity;  but  when  x  exceeds  unity,  this  series  is  con- 
vergent, hence  it  is  always  available  for  computation. 

Again,     '!^':'Jl  =  _i_,  =  ,+.'  +  .-H  ^-^  +  ... , 
'^  dx  I  —  X 

and  hence  tanh"'  x  =  x -\- -  x' -\- -x' -\-  ~x' -{-... ,     (34) 

3  5  7 

From  (32),  (33),  (34)  are  derived  : 
.-1  I 


:c\\~'  X  =  coslr 


2,2       2.4.4       2 .4.6.6 


(35) 


LOGARITHMIC  EXPRESSION  OF  ANTI-FUNCTIONS.  27 

csch-'^  =  sinh-i  =  l-i-i-3  +  i.l-L-i3i_L  +  ..., 

X         X         2   TyX  2     \  ^X"         246  7^' 

^      ^2.2       2.  4.4^2.4.6.6         •    '   ^-5  ^ 

coth-'  .r  =  tanh-'  l  =  i4--L-fJ_4.  _L  + (77) 

X      X  ^  ix'  ^  t^x-  ^  7x'  ^  ^^^^ 

Prob.  44.  Show  that  the  series  for  tanh" Vx",  coth~*  .r,  sech~*  jr, 
are  always  available  for  computation. 

Prob.  45.  Show  that  one  or  other  of  the  two  developments  of  the 
inverse  hyperbolic  cosecant  is  available. 

Art.  19.     Logarithmic   Expression  of  Anti  Fun'ctions. 


Let  X  =  cosh  //,      then  Vx''  —  i  =  siiih  u\ 


therefore  x  -\-  Vx^  —  i  =  cosh  u  -\-  sinh  //  =  e", 

and  u,  =  cosh"*.r,  =  log  (x  -|-  \^x''  —  i).  (38) 

Similarly,    sinli"*^  =  log  (^x  -\-  Vx''  -\-  i).  (39) 


Also  sech"'.r  =  cosh"*-  =  log  — — -,  (40) 


1-1            •  t  -ii         I        I  -{-  V\  A-  x^  /     , 

csch  ^x  =  sinh    -  =  log  — ■ •.  (41) 

X  X 

Again,  let  x  =  tanh  u  = 


^"  +  ^-"' 


therefore       — —  =  -^  =  e  , 


I  —  X      e' 


2u  =.\o%-^^ — ,     tanh  ^x^^\  log     "*"     ;        (42) 

I  X  -A-  I 

and  coth~';f  =  tanh"'-  =  \  log  — ! — .  (43) 

X  X  —   I 

Prob.  46.   Show  from  (38),  (39),  that,  when  .r^  00, 

sinh~'jc  —  log  :V:^  log  2,  cosh"'jc    -  log  x -i  log  2, 

and  hence  show  that  the  integration-constants  in  (32),  {^^)  are  each 
equal  to  log  2. 


28  HYPERBOLIC    FUNCTIONS. 

Prob.  47.  Derive  from  (42)  the  series  for  tanh    '.v  given   in  (34). 
Prob.  48.  Prove  the  identities: 

logA-  =  2  tanh"'" =:tanh'*  — —  =sinh'M(j:— jc"')=cosh"H(A-  +  A-"'): 

a-  + 1  x'  +  i  "  "  ^  " 

log  sec  .r  =  2  tanh"'  tan"  ^x;  log  esc  x  =  2  tanh'  '  tan'(j7r  -|-  -kx); 

log  tan  .r  =  —  tanh''  cos  2X  =  —  sinh"'  cot  2x  =  cosh"'  esc  2X. 

Art.  20.  The  Gudermanian  Function. 

The  coirespondence  of  sectors  of  the  same  species  was  dis- 
cussed in  Arts.  1-4.  It  is  now  convenient  to  treat  of  the 
correspondence  that  may  exist  between  sectors  of  different 
species. 

Two  points  P^,P^,  on  any  h)-perbola  and  enipse,are  said  to 
correspond  with  reference  to  two  pairs  of  conjugates  O^A^, 
O^B^  ,  and  O^A^,  O^B^,  respectively,  when 

-t'i/^>  =  ^^/'^'.>  (44) 

and  when  J,  jjj/j  have  the  same  sign.  The  sectors  A^O,P., 
A^O^P^  are  then  also  said  to  correspond.  Thus  corresponding 
sectors  of  central  conies  of  different  species  are  of  the  same 
sign  and  have  their  primary  characteristic  ratios  reciprocal. 
Hence  there  is  a  fixed  functional  relation  between  their  re- 
spective measures.  The  elliptic  sectorial  measure  is  called 
the  gudermanian  of  the  corresponding  hyperbolic  sectorial 
measure,  and  the  latter  the  anti-gudermanian  of  the  former. 
This  relation  is  expressed  by 

SJK,  =  gd  SJK, 

or     z>  =■  gd  //,     and     11  =  gd"'t'.  (45) 

Art.  21.    Circular  Functions  of  Gudermanian. 

The  six  hyperbolic  functions  of  11  are  expressible  in  terms 
of  the  six  circular  functions  of  its  gudermanian  ;  for  since 

—  =  cosh  u,        —  =  cos  7',  (see  Arts.  6,  7) 

in  which  //,  t-  are  the  measures  of  corresponding  h)-perbolic 
and  elliptic  sectors, 


hence 


GUDERMANtAN    ANGL£. 

cosh  u  =  sec  z>,  [eq.  (44)] 

sinh  H  =  v'secV  —  i  =  tan  7', 


29 


(46) 


/ 


tanh  u  =  tan  t'/sec  v  =  sin  v, 

COth  H   =  CSC  v, 

sech  //  =  COS7', 

csch  u  =  cot  ^'. 

The  gudei-maiiian  is  sometimes  useful  in  computation  ;  for 
instance,  if  sinh  u  be  given,  i'  can  be  found  from  a  table  of 
natural  tarigents,  and  the  other  circular  functions  of  z'  will  give 
the  remaining  hxperbolic  functions  of  //.  Other  uses  of  this 
function  are  given  in  Arts.  22-26,  32-36. 

Prob.  49.   Prove  that  gd  u  —  sec~'(cos]i  ti)  =  tan" '(sinh  u)  ^ 

=  COS"" '(sech  u)  =  sin" '(tanh  u), 

Prob.  50.  Prove   gd  "' Z'  =  cosh" '(sec  z')   =  sinh"' (tan  z') 

=  sech"'(cos  z')  =   tanh" '(sin  ?'). 

Prob.  51.    Prove      gd  o  =  o,  gd  00  =  ^-;r,       gd(—  00 )  =  —  ^^r. 
gd"'  0=0,  gd~'(4;r)  =00,  gd   '(— ^/7)=  — 00. 

Prob  52.   Show  that  gd  //  and  gd" '  z'  are  odd  functions  of  //,  z'. 

Prob.  53.  From  the  first  identity  in  4,  Prob.  17,  derive  the  rela- 
tion tanh  iu  —  tan  ^z'.  J 

Prob.  54.  Prove 

tanh"  '(tan  //)  =  4  gd  21/,  and  tan" '(tanh  x)  =  4  gd"'2A-. 

Art.  22.     Gudermanian  Angle 

If  a  circle  be  used  instead  of  the  ellipse  of  Art.  20,  the 
gudermanian  of  the  hyperbolic  sectorial  measure  will  be  equal 
to  the  radian  measure  of  the  angle  of  the  corresponding  circular 
sector  (see  eq.  (6),  and  Art.  3,  Prob.  2).  This  angle  will  be 
called  the  gudermanian  angle  ;  but  the  gudermanian  function  z', 
as  above  defined,  is  merely  a  number,  or  ratio  ;  and  this  number 
is  equal  to  the  radian  measure  of  the  gudermanian  angle  6, 
which  is  itself  usually  tabulated  in  degree  measure  ;  thus 

6  =  i8o°t'/n- (47) 


:^o 


HYPERBOLIC    FONCTlONS. 


Prob.  55.  Show  that  the  gudermanian  angle  of//  may  be  construct- 
ed as  follows: 

Take  the  principal  radius  OA  of  an  equilateral  hyperbola,  as  the 
initial  line,  and  OP  as  the  terminal 
line,  of  the  sector  whose  measure  is  u\ 
from  M,  the  foot  of  the  ordinate  of 
P,  draw  MT  tangent  to  the  circle 
wliose  diameter  is  the  transverse  axis; 
then  A07'\?>  the  angle  required.* 

Prob.  56.     Show    that    the    angle    B 
never  exceeds  90°. 

Prob.  57.   The  bisector  of  angle  AOT 

bisects  the  sector  AOP  (see   Prob.   13, 

Art.  9,  and  Prob.  53,  Art.  21),  and  the  line  AP.      (See  Prob.  i,  Art.  3.) 

Prob.  58.   This   bisector  is   parallel  to  TP,  and  the  points  7', /* 

are  in  line  with  the  point  diametrically  opposite  to  A. 

Prob.  59.   The   tangent    at    ''  passes   through   the    foot    of   the 
oidinate  of  T,  and  intersects  TM  ow  the  tangent  at  A. 

Prob.  60.   The  angle  AP M  is  half  the  gudermanian  angle. 

Art.  23.     Derivatives  of  Gudermanian  and  Inverse. 
Let  V  =  gd  u,     71  ■=  gd~'  z/, 

then  sec  v  =  cosh  u, 

sec  V  tan  vdv  =  sinh  n  du, 
sec  7'di>  =  ////, 
therefore  ^(gd"'  7-)  =  sec  7>  dv.  (48) 

Again,  //t'  =  cos  ■:>  di/    =:  sech  71  dn, 

therefore  //(gd  7/)  —  sech  //  du.  (49) 

Prob.  61.   Differentiate: 

y  =  sinh  //  —  gd  //,  y  =  sin  ?•  +  gd~'  7>, 

y  =  tanh  //  sech  //  +  gd  //,      y  =  tan  7'  sec  v  -\-  gd~'  v. 

*This  angle  was  called  by  Gudermann  the  longitude  of  u.  and  denoted  by  lu. 
His  inverse  symbol  was  li, ;  thus  «  =  ILU")-  (Crelle's  Journal,  vol.  6,  1S30.) 
Lambert,  who  introduced  the  angle  5,  named  it  the  transcendent  angle.  (Hist, 
de  I'acad  roy  de  Kerlin,  1761).  Hoiiel  (Nouvelles  Annales,  vol.  3,  1864) 
called  it  the  hvperbolic  amplitude  of  //,  and  wrote  it  amh  n,  in  analogy  with  the 
amplitude  of  an  elliptic  function,  as  shown  in  Prob.  62.  Cayley  (Elliptic 
Functions.  1876)  made  the  usage  uniform  by  attaching  to  the  angle  the  name 
of  the  mathemaiician  who  had  used  it  extensively  in  tabulation  and  in  the 
theory  of  elliptic  functions  of  modulus  unity., 


SERIES  FOR  GUDERMANIAN   AND  ITS  INVERSE.  31 

Prob.  62.   Writing    the    "elliptic   integral   of   the   first   kind"  in 
the  form 


J        Vi  —  K^  sin'''  0' 


X"  being  called  the  modulus,  and  (p  the  amplitude;  that  is, 

0  =  am  //,  (mod.  k), 

show  that,  in  the  special  case  when  k  =  i, 

u  =  gd~^  0,  am  It  =  gd  u,      sin  am  u  =  tanh  «, 

cos  am  //  =  sech  ?/,       tan  am  //  =  sinh  //; 

and  that  thus  the  elliptic  functions  sin  am  //,  etc.,  degenerate  into 
the  hyperbolic  functions,  when  the  modulus  is  unity.* 

Art.  24.     Series  for  Guderm.\nian  and  its  Inverse. 

Substitute   for  sech  //,  sec  t'  in  (49),  (48)  their  expansions, 
Art.  16,  and  integrate,  then 

gd  ;/  =  ;/-  iu'  +  ^\u'  -  ^^^^//'  +  .  .  .  (50) 

gd-'z'  =  v  +  Iz-'  +  ^V^.^  +^tio^''  +  •  .  .  (51) 

No  constants  of  integration  appear,  since  gd  u  vanishes  with 
u,  and  gd'^z>  with  7>.  These  series  are  seldom  used  in  compu- 
tation, as  gd  u  is  best  found  and  tabulated  by  means  of  tables 
of  natural  tangents  and  hyperbolic  sines,  from  the  equation 

gd  !(  =  tan~'(sinh  n), 
and  a  table  of  the  direct  function  can  be  used  to  furnLsh  the 
numerical  values  of  the   inverse  function  ;  or  the  latter  can  be 
obtained  from  the  equation, 

gd"'z^  =  sinh  "'(tan  Z')  =  cosh~'(sec  z'). 
To  obtain  a  logarithmic  expression  for  gd"':',  let 
gd""'t^  =  u,   z'  =  gd  i(, 

*  The  relation  gd  u  —  am  u,  (mod.  i),  led  Hoiiel  to  name  the  function  gd  u, 
the  hyperbolic  amplitude  of  m,  and  to  write  it  amh  //  (see  note,  Art.  22).  In  this 
connection  Cayley  expressed  the  functions  tanh  «,  sech  u.  sinli  u  in  the  form 
sin  gd  u,  cos  gd  u.  tan  gd  u,  and  wrote  them  sg  «,  eg  u,  tg  tt,  to  correspond 
with  the  abbreviations  sn  u,  en  u,  dn  u  for  sin  am  it,  cos  am  «.  tan  am  u. 
Thus  tanh  «  =  sg  «  =  sn  u,  (mod.  i);  etc. 

It  is  well  to  note  that  neither  the  elliptic  nor  the  hyperbole  functions 
received  their  names  on  account  of  the  relation  existing  between  them  in  a 
special  case.     (See  foot-note,  p.   7  ) 


32 


HYPERBOLIC    FUNCTIONS. 


therefore  sec  v  =  cosli  ?/,     tan  v  =  sinh  u, 

sec  V  -f-  tan  v  =  cosh  u  -\-  sinh  u  =■  e", 
I  -\-  sin  V  _i  —  cos  (^;r  -|-  z/) 


e"  = 


cos  t^  sin  {^rr  -\-  v) 

21,  =  gd  'v,  =  log,  tan  (i^  +  |t'). 


tan  (iTT  +  ^j;), 


(52) 


Prob.  6^.   Evaluate 


gd  u  —  u 


gd  'z'  —  v' 


Prob.  64.   Prove  that  gd  u  —  sin  u  is  an  infinitesimal  of  the  fifth 
order,  when  //  =  o. 

Prob.  65.  Prove  the  relations 

Itt  +  h'=  tan"V',     i^r  —  h;  —  tan"V~". 

Art.  25.    Graphs  of  Hyperbolic  Functions. 

Drawing  two  rectangular  axes,  and  laying  down  a  series  of 
points  whose  abscissas  represent,  on  any  convenient  scale,  suc- 
cessive values  of  the  sectorial  measure,  and  whose  ordinates 
represent,  preferably  on 
the  same  scale,  the  corre- 
sponding values  of  the 
function  to  be  plotted,  the 
locus  traced  out  by  this 
series  of  points  will  be  a 
graphical  representation  of 
the  variation  of  the  func- 
tion as  the  sectorial  meas- 


GRAPHS    OF    THE    HYPERBOLIC    FUNCTIONS.  33 

ure  varies.     The  equations  of  the  curves  in  the  ordinary  carte- 
sian notation  are : 

Fig.  Full  Lines.  Dotted  Lines. 

A         y  =^  cosh  X,         y  =  sech  x  ; 

B         y  =  sinh  x,        y  =  csch  x  ; 

C         y  ^  tanh  x,        y  =  coth  x  ; 

D         ^  =  gd  X. 

Here  x  is  written  for  the  sectorial  measure  //,  and  j  for  the 
numerical  value  of  cosh  li,  etc.  It  is  thus  to  be  noted  that  the 
variables  x,  y  are  numbers,  or  ratios,  and  that  the  equation 
y  =  cosh  X  merely  expresses  that  the  relation  between  the 
numbers  x  and  j  is  taken  to  be  the  same  as  the  relation  be- 
tween a  sectorial  measure  and  its  characteristic  ratio.  The 
numerical  values  of  cosh  u,  sinh  u,  tanh  u  are  given  in  the 
tables  at  the  end  of  this  chapter  for  values  of  u  between  o  and 
4.  For  greater  values  they  may  be  computed  from  the  devel- 
opments of  Art.  16. 

The  curves  exhibit  graphically  the  relations : 
sech  u  =  — : —  ,      csch  7f  =  -— - — ,    coth  u 


cosh  !('  sinh  u  tanh  //' 

cosh  u  <  I,     sech  u  >  i,     tanh  //  >  i,     gd  ;/  <^;r,  etc. ; 
sinh  (—  !/)  =  —  sinh  u,     cosh  (—  //)  =  cosh  //, 
tanh  (—  «)  =  —  tanh  ?/,     gd  {—  ?/)    =  —  gd  //,  etc.; 
cosh  0=1,     sinh  0  =  0,     tanh  0  =  0,     csch  (o)  =00  ,  etc.; 
cosh  (i  00 )  =  CO  ,  sinh  (it  00 )  =  ^oo  ,  tanh  (±  00 )  =  ±  i,  etc. 

The  slope  of  the  curve  j'  =  sinh  x  is  given  by  the  equation 
dy/dx  =  cosh  x,  showing  that  it  is  always  positive,  and  that 
the  curve  becomes  more  nearly  vertical  as  x  becomes  infinite. 
Its  direction  of  curvature  is  obtained  from  d'^y/dx'^  —  sinh  x, 
proving  that  the  curve  is  concave  downward  when  x  is  nega- 
tive, and  upward  when  x  is  positive.  The  point  of  inflexion  is 
at  the  origin,  and  the  inflexional  tangent  bisects  the  angle 
between  the  axes. 


34 


HYPERBOLIC    FUNCTIONS. 


The  direction  of  curvature  of  the  locus  j  =  sech  x  is  given 
by  dy/dx'  —  sech  x{2  tanh'jr  —  i),  and  thus  the  curve  is  con- 
cave   downwards    or    upwards 
according   as  2  tanh' ^  —  i    is 
negative  or  positive.     The  in- 

""'~ flexions    occur    at    the    points 

X  =  ±  tanh-'.707,  =  ±  .881, 
y  =  .707  ;  and  the  slopes  of 
the  inflexional  tangents  are 
=Fi/2. 

The  curve  y  =  csch  x  is 
asymptotic  to  both  axes,  but 
approaches  the  axis  of  x  more 
rapidly  than  it  approaches  the 
axis  of  )',  for  when  x  :=  3,  j  is 
onh'  .1,  but  it  is  not  till  _j'  =  10 


-I - 


that  X  is  so   small  as  .T,     The  curves  j' 
cross  at  the  points  ^  =  ±  .881,  j  =  ±  i. 


csch  X,  y  =  sinh  x 


Prob.  66.  Find  the  direction  of  curvature,  the  inflexional  tan- 
gent, and  tlie  asymptotes  of  the  curves  jr  =  gd  .v,  v  —  tanh  .v. 

Prob.  67.  Show  that  there  is  no  inflexion-point  on  the  curves 
y  z=  cosh  X,  y  =  coth  x. 

Prob.  68.  Show  that  any  line  _v  =  mx  +  //  meets  the  curve 
y  =  tanh  x  in  either  three  real  points  or  one.  Hence  prove  that 
the  equation  tanh  x  =  f/ix  -(-  n  has  either  three  real  roots  or  one. 
From  the  figure  give  an  approximate  solution  of  the  equation 
tanh  .V  =  .V  —  i. 


ELEMENTARY  INTEGRALS.  35 

Prob.  69.  Solve  the  equations:    cosh  ;v  — ■  .v -(-  2;  sinh  x  =  ^x; 

gd  X  =  X  —  ^TT. 

Prob.  70.  Show  which  of  the  graphs  represent  even    functions, 
and  which  of  them  represent  odd  ones. 

Art.  26.     Elementary  Integrals. 

The  following   useful   indefinite  integrals  follow  from  Arts. 
14.  15.  23: 

Hyperbolic.  Circular. 

1.  /  sinh  It  du  =  cosh  //,  /  sin  11  dii  =  —  cos  u, 

2.  I  cosh  11  dti  =  sinh  u,  j  cos  ?/  du  =  sin  k, 

3.  /  tanh  u  du  =  log  cosh  u,  I  tan  u  du  =  —  log  cos  //, 

4.  /  coth  ;/  du  =  log  sinh  u,  j  cot  //  du  =  log  sin  u, 

5.  y^csch  udu  =  log  tanh  -  ,      /esc  u  du  =  log  tan  -, 

=  —  sinh-'(csch  //),  =  —  cosh-'(csc  u), 

6.  Aech  u  du  =  gd  u,  I  sec  u  du  =  gd-'  u, 

7.  /  =  sinh-  -/      ./      ,  ,  =  sin-'  -, 

r      dx  ,      X-  r     —dx  ,x 

8.  /      ,  =  cosh-'  - ,       / 


COS" 


Q.    /  -^ i  =-tanh-'-,   /  -,— — I  =-tan-  — , 

*  Forms  7-12  are  preferable  to  the  respective  logarithmic  expressions 
(Art.  19),  on  account  of  the  close  analogy  with  the  circular  forms,  and  also 
because  they  involve  functions  that  are  directly  tabulated.  This  advantage 
appears  more  clearly  in  13-20. 


36 


HYPERBOLIC    FUNCTIONS. 


lO. 


I  I. 


/—dx  ~|  I        ^      X      r  —  dx  I  .X 

~ i  =-coth-'-,   / 


=  —  cot"'  — 
a  ^   a  -\-  X  a  a 


12 


r      —  dx  I        ,     ,  t'     /*        dx  I 

/   —  —  — sccn~  —     /   —  ^i:  — 

«/  ;r  4/^^  _  x'      a  a'  ^   X  s/x"  -a'       a 

X      P      —  dx  I 


see" 


./ 


=  -  csch-' 


X  Vd'  -\-  x'       a 


a 

=  —  csc~  — , 
a'  ^   X  \ '  x^  -  d^      a  a 


From   these    fundamental   integrals  the  following  may  be 
derived : 


.3./ 


dx 


I                  ax  -\-  b 
=  — ^  sinh~   —- ,  tf  positive,  tfr>  3"; 


Vax'  -j-2dx-\-c       Va  Vac-  d' 


I          ,     ,   ax  4-/7 
=  — =cosh         ^         — ,  ^?  positive,  ^f<g; 
Va  \b''  —  ac 

I  ax-{-  b 

=  COS"  —  ,  a  negative. 


14, 


J  ax' 


dx 


V—a 
I 


tan 


\l)'  —  ac 
_.  nx-\-  b 


^2bx-\-c  Vac-b-"  V^t^^ 


,     ac>  b'; 


Vb — ac  Vb  —ac  ' 

—  I  ,    ax  4-  b  ,  

coth-'     ,^--^,  ac  <  /;^  .?a-  +  ^^  >  Vb'  -  ac  ; 


=  — coth--'(^-— 2) 


Vb'-ac 

Inus,    /  — 1 j— 

',./    ^'-4^-4-3 

=  tanh-'(.5)-tanh-'(.3333)  =  . 5494-. 3466=. 2028,* 


:coth-'2  — coth-'3 


/ -,    ^'^   ,      =-tanh-'(x-2)    =tanh-'o-tanh-'(.5) 
t/g  ;ir  —/\x-\-l  A-i 

=  -  •5494- 

(By  interpreting  these  two  integrals  as  areas,  show  graph- 

ically  that  the  first  is  positive,  and  the  second  negative.) 


5.     C '  "^ , =  —^^^^  tanh-'  K  /- 

J  {a-x\Vx-b       Va-b  V' 


a-F 


*For  tanh-' (.5)  interpolate  between  tanh  (.54)  =  .4930,  tanh  (.56)  =  .5080 
(see  tables,  pp.  6^,  65);  and  similarly  for  tanh-'  (.3333). 


ELEMENTARY    INTEGRALS.  37 


/  '^-^  2  Ix  —  b 

tan  ~    \      —, ,  or   — ,  cotn~ 


\'b-a  V    ^'-(^  Va-b  V  '^-f^ 

the  real  form  to  be  taken.     (Put  x  —  b  =  s",  and  apply  9,  10.) 


^      r  dx  2  b—x 

16.     /; --^==:— ==tanh-'A  /t 

t/   {a—x\  Vb—x 


{a—x)\U-)—x       \'b  —  a  \l  b-a' 

2  j  b  —  x  —2  Ib—x 

or          ,-  coth"    A  /  -, ,   or    — —    -  tan  "'  a  /  ; ; 

\/b—a  V    '^'-^  \^a-b  V  ^~^ 

the  real  form  to  be  taken. 

(.1-'  —  rt-')-^/,;-  =  --t'(^r  —  <'?')^ ^/'  cosh-'-. 

By  means  of  a  reduction-formula  this  integral  is  easily  made 
to  depend  on  8.  It  may  also  be  obtained  by  transforming 
the  expression  into  hyperbolic  functions  by  the  assumption 
X  =  a  cosh  u,  when  the  integral  takes  the  form 

rt^  /  sinh'  udu=z  —  /  (cosh  2u  —  \)du  =  -f^^(sinh  2u  —  211) 

=  |rt'(sinh  u  cosh  u  —  ii), 

which  gives  17  on  replacing  a  cosh  71  by  .r,  and  a  sinh  u  by 
(,t"'  —  rt^)i.  The  geometrical  interpretation  of  the  result  is 
evident,  as  it  expresses  that  the  area  of  a  rectangular-hyper- 
bolic segment  AMP  is  the  difference  between  a  triangle  OMP 
and  a  sector  OAP. 

18.  J^{a'  -  .x'fdx  =  ~x{a'  -  x'f  +  ~a'  sin"'  -. 

19.  fix'  +  a'fdx  =  ^.r(.r'  +  a^f  -f  -^a^  sinh"'  ^. 

20.  /  sec'  0^/0  =  /  (I  -[-  tan-  0)v/  tan  0 

=  ^  tan  0(1  4-  tan'  0)^  -\-  ^  sinh"'  (tan  0) 
=  ^  sec  0  tan  0+2  gd"'  0. 

21.  /  sech'?/<//^=:  ^  sech  ?/  tanh  ?/  -|-  i  gd  ?^ 

Prob.  71.   What  is  the  geometrical  interpretation  of  18,  19? 
Prob.  72.  Show  that  /  («.v'  +  2kv  -j-  (r)W  reduces  to  17,  18,  19, 


38  HYPERBOLIC    FUNCTIONS. 

respectively:  when  a  is  positive,  with  ac  <  b^  ;  when  a  is  negative; 
and  when  a  is  positive,  with  ac  >  b\ 

Prob.  73.  Prove    /  sinh  u  tanh  //  du  —  sinh  u  —  gd  //, 


J 


t  2/ 

cosh  u  coth  //  (/u  =  cosh  u  -\-  log  tanh  — . 

2 


Prob.  74.  Integrate 
(a-'  +  2-v  +  5)-W,     (a-'^  +  2.v  +  5)-VA-,     Cv^  +  2x  +  s)V>. 

Prob.  75.  In  the  parabola  ^  =  4px,  if  j-  be  the  length  of  arc 
measured  from  the  vertex,  and  (p  the  angle  which  the  tangent  line 
makes  with  the  vertical  tangent,  prove  that  the  intrinsic  equation  of 
the  curve  is  ds/d<p  —  2/)  sec"  cp,  s  ^  p  sec  0  tan  0  +/gd~'0. 

Prob.  76.  The  polar  equation  of  a  parabola  being  ;■  =  a  sec'|^, 
referred  to  its  focus  as  pole,  express  s  in  terms  of  6*. 

Prob.  77.  Find  the  intrinsic  equation  of  the  curve  j/a  =  cosh  x/a, 
and  of  the  curve  j'/<z  =  log  sec  x/a. 

Prob.   78.   Investigate  a  formula  of  reduction  for  /  cos\\"xdx; 

also  integrate  by  parts  cosh'"'.T  (^/Ir,  tanh"'aw/A-,  (sinh"' a)\Zv;    and 

show  that  the  ordinary  methods  of  reduction  for    /  cos"'A"sin"Afl'.x 

can  be  applied  to  /  cosh'"  .r  sinh"  x  dx. 

Art.  27.     Functions  of  Complex  Numbers. 

As  vector  quatitities  are  of  frequent  occurence  in  Mathe- 
matical Physics;  and  as  the  numerical  measure  of  a  vector 
in  terms  of  a  standard  vector  is  a  complex  number  of  the 
.orm  A- -\~  I'j',  in  which  x,  j  are  real,  and  i  stands  for  V —  i;  it 
becomes  necessary  in  treating  of  any  class  of  functional  oper- 
ations to  consider  the  meaning  of  these  operations  when  per- 
formed on  such  generalized  numbers.*  The  geometrical  defini- 
tions of  cosh  7/,  sinh?/,  given  in  Art.  7,  being  then  no  longer 
applicable,  it   is  necessary  to    assign   to  each   of  the  symbols 

*The  11- e  of  vectors  in  electrical  theory  is  shown  in  Bedell  and  Crehore's 
Alternating  Currents,  Chaps,  xiv-xx  (first  published  in  1892).  The  advantage 
of  introducing  the  complex  measures  of  such  vectors  into  the  differential  equa- 
tions is  fhovvn  by  Steinmetz,  Proc.  Elec.  Congress,  1893;  while  the  additional 
convenience  of  expressing  the  solution  in  hyperbolic  functions  of  these  complex 
numbers  is  exemplified  by  Kennelly,  Proc.  American  Institute  Electrical 
Engineers,  April  1895.      (See  below,  Art.  37.) 


FUN'CTIONS    OF    COMPLEX    NUMBERS. 


39 


cosh  (^  -f-  //),  sinh  (.f -|-  tj'),  a  suitable  algebraic  meaning, 
which  should  be  consistent  with  the  known  algebraic  values  of 
cosh  ^,  sinh  ^,  and  include  these  values  as  a  particular  case 
when  _>/ =  O.  The  meanings  assigned  should  also,  if  possible, 
be  such  as  to  permit  the  addition-formulas  of  Art.  1 1  to  be 
made  general,  with  all  the  consequences  that  flow  from  them. 

Such  definitions  are  furnished  by  the  algebraic  develop- 
ments in  Art.  i6,  which  are  convergent  for  all  values  of  u,  real 
or  complex.  Thus  the  definitions  of  cosh  {x  -\-  ij),  sinh  [x  -f-  iy) 
are  to  be 


cosh  {x  +  iy)  =  I  +  ^{x  +  tyy  +  l-(x  +  iyy  + 

2  !  4  • 


sinh  {x  +  /»  =  {x  +  iy)  -f  -(.r  +  /jf  + 


(52) 


From  these  series  the  numerical  values  of  cosh  {x -{- iy), 
sinh  {x-\-iy)  could  be  computed  to  any  degree  of  approxima- 
tion, when  X  and  7  are  given.  In  general  the  results  will  come 
out  in  the  complex  form* 

cosh  {x  -f-  iy)  =  a-\-  ib, 
sinh  (.V  -|-  iy)  =  c  -\-  id. 
The  other  functions  are  defined  as  in  Art.  7,  eq.  (9). 

Prob.  79.   Prove  from  these  definitions  that,  whatever  u  may  be, 
cosh  (—//)  =  cosh  u,  ■  sinh  (—//)=—  sinh  «,    , 


lilt 


cosh  //     =  sinh  //, 


du 


sinh  u     =  cosh  u, 


7  2  72 

^^cosh  mil  =  ;;/' cosh  w//,   j-^  sinh  w//  =  w'  sinli  /////.f 


du 


du' 


*It  is  to  be  borne  in  mind  that  the  symbols  cosh,  sinh,  here  stand  for  alge- 
braic operators  which  convert  one  number  into  another;  or  which,  in  the  lan- 
guage of  vector-analysis,  change  one  vector  into  another,  by  stretching  and 
turning. 

f  The  generalized  hyperbolic  functions  usually  present  themselves  in  Mathe- 
matical Physics  as  the  solution  of  the  differential  equation  d''(p/dn'^  =  fi^<p, 
where  </>,  w,  u  are  complex  numbers,  the  measures  of  vector  quantities.  (See 
Art.  37.) 


40  HYPERBOLIC    FUNCTIONS. 

Art.  28.    Addition-Theorems  eor  Complexes. 

The  addition-theorems  for  cosh  {/i  -\-  7'),  etc.,  where  7i,  v  are 
complex  numbers,  may  be  derived  as  follows.  First  take  u,v 
as  real  numbers,  then,  by  Art.  Ii, 

cosh  {h  -\-  v)  —  cosh  7c  cosh  v -\-  sinh  u  sinh  v, 
hence    I  +  ^',(»  +  r)'  +,  ..=(.+  ^W  +  ...)(.  +  ^^'+. .  .) 

+  („  +  _L^„.  +  ...)(„+±y+...) 

This  equation  is  true  when  n,  v  are  any  real  numbers.  It 
must,  then,  be  an  algebraic  identity.  For,  compare  the  terms 
of  the  rt\\  degree  in  the  letters  //,  z'  on  each  side.     Those  on 

the  left  are  — (/^-|-  t')';  and  those  on  the  right,  when   collected, 

form  an  rth-degree  function  which  is  numerically  equal  to  the 
former  for  more  than  r  values  of  //  when  v  is  constant,  and  for 
more  than  r  values  of  v  when  u  is  constant.  Hence  the  terms 
of  the  rth  degree  on  each  side  are  algebraically  identical  func- 
tions of  //  and  z'.*  Similarly  for  the  terms  of  any  other  degree. 
Thus  the  equation  above  written  is  an  algebraic  identity,  and 
is  true  for  all  values  of  u,  v,  whether  real  or  complex.  Then 
writing  for  each  side  its  symbol,  it  follows  that 

cosh  {u  -\-  7')  =  cosh  ;/  cosh  7'  -|-  sinh  ii  sinh  v\  (53) 
and  by  changing  7'  into  —  7', 

cosh  {h  —  7')  —  cosh  //  cosh  v  —  sinh  //  sinh  7'.  (54) 

In  a  similar  manner  is  found 

sinh  {u  ±  1')  =  sinh  u  cosh  v  ±  cosh  71  sinh  v.  (55) 

In  particular,  for  a  complex  argument, 

cosh  (x  ±  ij)  =  cosh  x  cosh  ij'  ±  sinh  x  sinh  /)',  ) 

[       (56) 
sinh  {x  ±  /r)  =  sinh  x  cosh  ly  ±  cosh  x  sinh  ?j'.  ) 

*  "  If  two  ;'lh-degree  functions  of  a  single  variable  be  equal  for  more  than  r 
values  of  the  variable,  then  they  are  equal  for  all  values  of  the  variable,  and  are 
algebraically  identical." 


fUNCTlONS    OF    PURE    IMAGINARIES.  41 

Prob.  79.  Show,  by  a  similar  process  of  generalization,*  that  if 
sin  //■,  cos  //,  exp  ti  \  be  defined  by  their  developments  in  powers  of 
ti,  then,  whatever  u  may  be, 

sin  (//  -\-  v)  ^^  sin  u  cos  v  +  cos  u  sin  z', 
cos  (//  -|-  ^')  =  cos  //  cos  V  —  sin  //  sin  v,    ^-"^ 
exp  (/^  -\-  7')  =  exp  //  exp  ?'. 
Prob.  80.   Prove  that  the  following  are  identities: 
cosh''  //  —  sinh*  /^  =  i, 
cosh  //  -f-  sinh  //  =  exp  //, 
cosh  //  —  sinh  u  =  exp  (  —  //), 
cosh  //  =  o[exp  //  4"  tx]:)  (  —  //)], 
sinh  //   =  i[exp  //  —  ex])(—  //)]. 

Art.  29.    Functions  of  Pure  Imaginaries. 

In  the  defining  identities 

cosh  ?(  =  !-[-  ~ii^  A -//*  -I-  .  .  ., 

2!  4!        '  ' 

sinh  11  ■=  21  -\ — -^11'  -J —  /'^  -f-  .  .  ., 
3-  5- 

put  for  //  the  pure  iniaginary  //,  then 

cosh  iy  ^  \  —  --/  -I-  -  /  -  .  .  .  =  COS7,  (57) 

z.  4 

sinh  iy  =  iy  ^  -,(/»'  +  -,(?»'  +  .  .  . 


{/-^y+^/ 


=  /sin/,      (58) 


and,  by  division,       tanh  iy  =  /  tan  y.  (59) 

*  This  method  of  generalization  is  sometimes  called  the  principle  of  the 
"  permanence  of  equivalence  of  forms."  It  is  not,  however,  strictly  speaiving,  a 
"  priiiciple,"  but  a  method;  for,  the  validity  of  the  generalization  has  to  be 
demonstrated,  for  any  particular  form,  by  means  of  the  principle  of  the  alge- 
braic identity  of  polynomials  enunciated  in  the  preceding  foot-note.  (See 
Annals  of  Mathematics,  Vol.  6,  p.  81.) 

f  The  symbol  exp  u  stands  for  "exponential  function  of  u,"  which  is  identi- 
cal with  e'<  when  it  is  real. 


4^ 


HYPERBOLIC    FUNCTIONS. 


These  formulas  serve  to  interchange  hyperbohc  and  circular 
functions.  The  hyperbolic  cosine  of  a  pure  imaginary  is  real, 
and  the  ii\-perbolic  sine  and  tangent  are  pure  imaginaries. 

The  following  table  exhibits  the  variation  of  sinh  u,  cosh  u, 
tanh  II,  exp  u,  as  u  takes  a  succession  of  pure  imaginary  values. 


tt 

sinh  u 

cosh  u 

tanh  u 

exp  u 

o 

O 

I 

0 

I 

\^^ 

.yi 

.7* 

i 

•7(1+0 

^171 

i 

0 

CO    / 

i 

lirr 

.7/ 

-•7 

—  i 

•7(1  -  i) 

0 

—  I 

0 

—  I 

\iTt 

-.ji 

-.7 

i 

-.7(1+0 
—  i 

pTT 

—  i 

0 

00  /' 

liTt 

-.7i 

•7 

—  i 

-.7(1-0 

2iit 

0 

1 

0 

I 

*  In  this  table  .7  is  written  for  \  \/i,  =  .707  .  . 
Prob.  81.   Prove  the  following  identities  : 

cos  J  =  cosh  /)■      =  i>[exp  /)■  +  exp  (—  /v)], 

sin  J'  =  -  sinh  /)'  =     -[exp  iy  —  exf)  (—  /v)], 

cos  }'  -\-  i  sin  y  =  cosh  iy  +  sinh  /)'  =  exp  iy, 
cos^y  —  /  sin_)'  =  cosh  iy  —  sinh  iy  =  exp  (—  ^V), 
cos  iy  =  cosh  y,     sin  iy  =  i  sinh  v. 

Prob.  82  Equating  the  respective  real  and  imaginary  parts  on 
each  side  of  the  equation  cos  ny  -f  i  sin  f/y  =  (cos  y  +  i  sin  _>')", 
express  cos  //y  in  powers  of  cos_v,  sin  v  ;  and  lience  derive  the  cor- 
responding expression  for  cosh  ny. 

Prol).  83.  SIiow  that,  in  the  identities  (57)  and  (58),  y  may  be 
replaced  by  a  general  complex,  and  hence  that 

sinh  (x  ±  iy)  =  ±  i  sin  {y  T  /v), 


FUNCTIONS  OF  .V  -f  /)'  IN   THE  FORM   A'  -(-  I  V.  43 

cosh  (.v  ±  iv)  =  COS  (  v  ^  is), 
sin  {x  ±  /)■)  =  ±  /sinh  (r  ^  ix), 
cos  (.r  ±  /V)  =  cosh  ( )'  =F  ix). 

Prob.  84.  From    the    product-series    for  sin  .v    derive   that    for 
sinh  X  : 

(     ^t^v      ^'-^  V      ■^■ 

sin  vT  =  a-  I r,     I — — ,   II  — 


7f/\         2'n'')\  T^'n 

Art.  30.     Functions  of  x  ^  iy  in  the  Form  X -[- iY. 
By  the  addition-formulas, 

cosh  (.r  -(-  iy)  =  cosh  x  cosh  iy  -\-  sinh  x  sinh  iy, 
sinh  (,t'  -j-  /j)  =  sinh  x  cosh  z/  -f-  cosh  ,r  sinh  z/, 
but  cosh  iy  =  cos  y,      sinh  iy  =  /  sin  y, 

hence    cosh  {x  -\-  iy)  =  cosh  x  cos  y  -\-  i  sinh  x  sin  y, 


^n 


...  .  .         (60) 

sinh  (x  -\-  iy)  =  sinh  x  cos  y  -|- 1  cosh  ,t'  sin  j. 

Thus  if  cosh  (x -\-  iy)  =  a-}- id,  sinh  {x  -\-  iy)  =  c  -\-  it/,  then 

a  =  cosh  X  cos  ;/,     /?  =  sinh  ,r  sin  y, 


(61) 
^  =  sinh  X  cos  jj/,     ^/  =  cosh  x  sin  j' 

From  these  expressions  the  complex  tables  at  the  end  of 
this  chapter  have  been  computed. 

Writing  cosh  s  =Z,  where  ::  =  x  -^  iy,  Z  =  XA^  iV;  let  the 
complex  numbers  s,  Z  he  represented  on  Argand  diagrams,  in 
the  usual  way,  by  the  points  whose  coordinates  are  (x,  y), 
{X,  F);  and  let  the  point  z  move  parallel  to  the  j-axis,  on  a 
given  line  x  =  ;//,  then  the  point  Z  will  describe  an  ellipGe 
whose  equation,  obtained  by  eliminating  y  between  the  equa- 
tions X  =^  cosh  ;//  cos  y,  Y=  sinh  vi  sin  y,  is 
X'  V 


(cosh  my       (sinh  mf 

and  which,  as  the  parameter  m  varies,  represents  a  series  of 
confocal  ellipses,  the   distance  between  whose    foci  is   unity. 


44'  HYPERBOLIC    FUNCTIONS. 

Similarly,  if  the  point  z  move  parallel  to  the  ;tr-axis,  on  a  given 
line  J  =  «,  the  point  Z  will  describe  an  liyperbola  whose  equa- 
tion, obtained  by  eliminating  the  variable  x  from  the  equations 
A'=  cosh  X  cos  ;/,  Y  =  sinh  x  sin  n,  is 

_JC^ F^  _ 

(cos  //)'        (sin  ny 

and  which,  as  the  parameter  n  varies,  represents  a  series  uf 
hyperbolas  con  focal  with  the  former  series  of  ellipses. 

These  two  systems  of  curves,  when  accurately  drawn  at 
close  intervals  on  the  Z  plane,  constitute  a  chart  of  the  hyper- 
bolic cosine;  and  the  numerical  value  of  cosh  (;//  -j-  /;/)  can  be- 
read  off  at  the  intersection  of  the  ellipse  whose  parameter  is  vi 
with  the  hyperbola  whose  parameter  is  «.*  A  similar  chart  can 
be  draw^n  for  sinh   {x-\-iy),   as  indicated  in  Prob.  85. 

Periodicity  of  Hyperbolic  Functions. — The  functions  sinh  m 
and  cosh  u  have  the  pure  imaginary  period  2/-.     For 

sinh  (M  +  2/;r)  =sinh  u  cos  27r  +  ?  cosh  u  sin  27:  =  sinh  w, 
cosh  {u\2iTi)  =cosh  u  cos  2ti-\-i  sinh  u  sin  2;:  =  cosh  w. 
The  functions  sinh  u  and  cosh  u  each  change  sign  when  the 
argument  u  is  increased  by  the  half  period  irr.     For 

sinh  (w  +  /r:)  =sinli  u  cos  ;:  +  i  cosh  w  sin  ;:=  —sinh  w, 
hd  tt '  cosh  («  +  /;:)=  cosh  u  cos  7r  +  i  sinh  w  sin  ;r=  —cosh  u. 

The  function  tanh  u  has  the  period  iit.     For,  it  follows  from 
the  last  two  identities,   by  dividing  member  by  member,   that 
tanh  {u-^iTz)  =tanh  u. 
By  a  similar  use  of  the  addition  formulas  it  is  shown  that 

sinh  {u\\iiz)  =i  cosh  u,     cosh  {u  +  ^ir:)  =i  sinh  u. 
By  means  of  these  periodic,  half-periodic,  and  quarter-periodic 
relations,  the  hyperbolic  functions  of  x-\-iy  are  easily  expressible 
in  terms  of  functions  of  x  -f  iy',  in  which  y'  is  less  than  ^iz. 

*  Such  a  chart  is  given  by  Kennelly,  Proc.  A.  I.  E.  E.,  April  1895,  and  is 
used  by  him  to  obtain  the  numerical  values  of  cosh  {x -\- iy)  sinh  (.r-|- (r),  which 
present  themselves  as  the  measures  of  certain  vector  quantities  in  the  theory  of 
alternating  currents.  (See  Art.  37.)  The  chart  is  constructed  for  values  of  x 
and  of  1'  between  o  and  1.2;  but  it  is  available  for  all  values  of  r,  on  account  of 
the  periodicity  of  the  functions. 


FUNCTIONS    OF    xi-iy   IN    THE    FORM    X+IY.  45 

The  hyperbolic  functions  are  classed  in  the  modern  function- 
theory  of  a  complex  variable  as  functions  that  are  singly  periodic 
with  a  pure  imaginary  period,  just  as  the  circular  functions  are 
singly  periodic  with  a  real  period,  and  the  elliptic  functions  are 
doubly  periodic  with  both  a  real  and  a  pure  imaginary  period. 

Multiple  Values  of  Inverse  HyperboHc  Functions. — It  fol- 
lows from  the  periodicity  of  the  direct  functions  that  the  inverse 
functions  sinh~^  m  and  cosh~i  m  have  each  an  indefinite  number 
of  values  arranged  in  a  series  at  intervals  of  2/;r.  That  partic- 
ular value  of  sinh~^w  which  has  the  coefficient  of  i  not  greater 
than  |7r  nor  less  than  —^n  is  called  the  principal  value  of  sinh~i  w; 
and  that  particular  value  of  cosh"^  m  which  has  the  coefficient  of  i 
not  greater  than  n  nor  less  than  zero  is  called  the  principal  value 
of  cosh~^w.  When  it  is  necessary  to  distinguish  between  the 
general  value  and  the  principal  value  the  symbol  of  the  former 
will  be  capitalized ;   thus 

Sinh~i  m   =  sinh~^  m  +  2ir7i,       Cosh~^  m  =  cosh~^  m  +  2/>7r, 
Tanh~^  m  =  tanh~i  m  +  irrc, 
in  which  r  is  any  integer,  positive  or  negative. 

Complex  Roots  of  Cubic  Equations. — It  is  well  known  that 
when  the  roots  of  a  cubic  equation  are  all  real  they  are  expressible 
in  terms  of  circular  functions.  Analogous  hyperbolic  expressions 
are  easily  found  when  two  of  the  roots  are  complex.  Let  the 
cubic,  with  second  term  removed,  be  written 

X^±7,bx=2C. 

Consider  first  the  case  in  which  b  has  the  positive  sign.  Let 
x  =  r  sinh  u,  substitute,  and  divide  by  r^,  then 

.    ,  ,         3^    .   ,  2C 

smh   u  +  ~  smh  u-^-t,. 
r^  r 


gives 


Comparison  with  the  formula  s!nh^  7/+f  sinh  u  =  \  sinh  3W 
3^     3       2C    sinh  2>u 


11-1  ^  I  ^ 

whence  r=20*,     smh3w  =  -T^,     w  =  -smh~^Tg; 

/  I    .  c 

therefore  x=2h^  sinh    -  sinh~^T5 

\3  b' 


46  HYPERBOLIC    FUNCTIONS. 

in  which  the  sign  of  b^  is  to  be  taken  the  same  as  the  sign  of  c. 

Now  let  the  principal  vakic  of  sinh^^Ty,  found  from  the  tables, 

be  n;    then  two  of    the  imaginary  values  are  n±2i~,  hence  the 

three   values   of   x  are  20-  smh  -  and   20-  sm..(-±  —  ).     The 

3  \3       3  / 

last  two  reduce  to  — /;Msinh  —  ±/\'^3  cosh  -j. 

Next,  let  the  coeflicient  of  .v  be  negative  and  equal  to  —T,b. 

It  miy  then  be  shown   similarly  that  the  substitution  x  =  r  sin  d 

leads  to  the  three  solutions 

,,    .    «       ,,  /  .     n       /—       w\         ,  c 

—  20*  sin-,      Ml  sm -±v  3  cos-J,     where  »  =  sm~^  rr. 

These  roots  are   all    real  when  f"%&-.     If  c>b^,    the  substitution 
x  =  rcosh7<  leads  to  the  solution 

:v  =  2&*cosh  (-cosh~iTyj, 

which  gives  the  three  roots 

ft  /  It  f7\  ^ 

2^  cosh  -,    —  /'■  ( cosh  -  ±  / V  3^  sinh  -  I ,   '.vherein  n  =  cosh"'*  tj , 
3  \  3  '  3/  b' 

in  which  the  sign  of  b^  is  to  be  taken  the  same  as  the  sign  of  c. 

Prob.  85.  Show  that  the  chart  of  cosh  (.r  +  ;))  can  he  adapted 

to  sinh  {x  -f-  /v),  by  turning  through  a  right  angle;  also  to  sin  (.v  +/V)- 

,     „,     ^  ,      .,       •  ,    /      ,     -s        sinli  2  ^ "+ '  sin  2r 

Prob.  80.   Prove  the  identity  tanli  (.v  -f-  t\)  =  ; '- . 

cosli  2.V  -j-  cos  2J' 

Prob.  87.   If  cosh  (x  -\-  iv),  =  a  -\-  ib,  be  written  ii'i  the  "  modulus 

and  amplitude"  form  as  r(cos  B  -\-  /sin  (^),  ~  r  exp  W,  then 

/-'  =  a"^  -\-  b'^  =1  cosh^  .V  —  sin^j'  =  cos'^'j'  —  sinh^  .r, 

tan  6  =  b/a  =  tanh  x  tan  7. 
Prob.  88.   Find  the  modulus  and  amplitude  of  sinh  {x  -\-  ty). 

Prob.  89.    Show  that  the  periotl  of  exp  is  id. 

a 

Prol).   90.    When    ;//    is    real    and   >  i,     cos~'  ffi   =  /  cosh~*  m, 

sin~'  ;//  =   —  —  /cosh    '  m. 
2 

When  m  is  real  and  <  i,  cosh"'  ;//  =  /  cos~'  m. 


THE    CATENARY.  4'^ 

Art.  31.    The  Catenary. 

A  flexible  inextensible  string  is  suspended  from  two  fixed 
points,  and  takes  up  a  position  of  equilibrium  under  the 
action  of  gravity.  It  is  required  to  find  the  equation  of  the 
curve  in  which  it  hangs. 

Let  w  be  the  weight  of  unit  length,  and  s  the  length  of  arc 
^/'measured  from  the  lowest  point  A  ;  then  zus  is  the  weight 
of  the  portion  AP.  This  is  balanced  by  the  terminal  tensions, 
T  acting  in  the  tangent  line  at  P,  and  H  in  the  horizontal 
tangent.     Resolving  horizontally  and  vertically  gives 

T  cos  (p  =  //,      T  s\n  (p  =  ws, 
in  which  0  is  the  inclination  of  the  tangent  at  P\  hence 

U'S        s 

tan0  =  ^=:-, 

wheie  c  is  written  for  ///7i>,  the   length  whose  weight  is  the 
constant  horizontal  tension  ;  therefore 


dy       s        lis  /         s"       dx  ds 


dx      c'      dx       Y        '    c"       c         \^s''  -f-  c""' 

X        .  ,    ,  -f      •  ,    '^        s       dy     y  x 

—  =  smh~  — ,  smh  —  =  —  =  3—,  —  =  cosh  -, 
c  c  c        c       dx      V  c 

which  is  the  required  equation  of  the  catenary,  referred  to  an 
axis  of  x  drawn  at  a  distance  c  below  A. 

The  following  trigonometric  method  illustrates  the  use  of 
the  gudermanian  :  The  "  intrinsic  equation,"  s  ^^  c  tan  0, 
gives  ds  =  c  sec''  0  <^/0;  hence  dx,  =  ds  cos  cp,  =  c  sec  (pd(p; 
dy,^=ds  sin  0,  =  r  sec  0  tan  0  d(p  ;  thus  x=c  gd"'  0,  y  =  c  sec  0; 
whence  y/c  =  sec  0  =  sec  gd  x/c  =  cosh  x/c ;  and 
s/c  =  tan  gd  x/c  =  sinh  x/c. 

Numerical  Exercise. — A  chain  whose  length  is  30  feet  is 
suspended  from  two  points  20  feet  apart  in  the  same  hori- 
zontal ;  find  the  parameter  c,  and  the  depth  of  the  lowest 
point. 


48  HYPERBOLIC    FUNCTIONS. 

The  equation  s/c  ■=■  sinh  x/c  gives  I'^/c  =  siiih  lo/c,  which, 
by  putting  lo/c  =  c,  may  be  written  i.5,c  =  sinh  ^.  By  exam- 
ining the  intersection  of  the  graphs  of;-  =  sinli;;,  y  =  1.5^, 
it  appears  that  the  root  of  this  equation  is  ;?  =  1.6,  nearly. 
To  find  a  closer  approximation  to  the  root,  write  the  equation 
in  the  iovm /[,a)  =  sinh  2  —  i.^^  =  o,  then,  by  tiie  tables, 

/(1.60)  =  2.3756  —  2.4000  =  —  .0244, 
/(1.62)  =  2.4276  —  2.4300  =:  —  .0024, 
/(1.64)  =  2.4806  —  2.4600  =  -f  -0206; 

whence,  by  interpolation,  it  is  found  that  y(i.622i)  =  o,  and 
z  =  1.622 1,  c  =  lo/s  =  6.1649.  The  ordinate  of  either  of 
the  fixed  points  is  given    by  the  equation 

j'/c  =  cosh  x/c  =  cosh  10/^  =  cosh  1.6221  =  2.6306, 

from  tables;  hence  j' =  16.2174,  and  required  depth  of  the 
vertex  =  j  —  r  =  10.0525  feet.* 

Prob.  91.  In  tlie  above  numerical  problem,  find  the  inclination 
of  the  terminal  tangent  to  the  horizon. 

Prob.  92.  If  a  perpendicular  AfJV  he  drawn  from  the  foot  of  the 
ordinate  to  the  tangent  at  P,  prove  that  A/iV  is  equal  to  the  con- 
stant r,  and  that  JVP  is  equal  to  the  arc  A  P.  Hence  show  that 
the  locus  of  JV  is  the  involute  of  the  catenary,  and  has  the  prop- 
erty that  the  length  of  the  tangent,  from  the  point  of  contact  to  the 
axis  of  .V,  is  constant.  (This  is  the  characteristic  property  of  the 
tractory). 

Prob.  93.  The  tension  Tat  any  point  is  ecjual  to  the  weight  of  a 
portion  of  the  string  whose  length  is  equal  to  the  ordinate  j'  of  that 
point. 

Prob.  94  An  nrch  in  the  form  of  an  inverted  catenary  f  is  30 
feet  wide  and  10  feet  higli;  show  that  the  length  of  the  arch  can  be 

obtained  from  the  ecp'.ations  cosh  5  —  — s  =1,       2S  ^=    "^    sinh  z. 

3  2 

*  See  a  similar  problem  in  Cha[).  I,  Art.  7. 

f  For  the  theory  of  this  form  of  arch,  sec   "Arch"  in  the  Encyclopaedia 
Britannica. 


CATENARY    OF    UNIFORM    STRENGTH.  49 

Art.  32.  Catenary  of  Uniform  Strength. 

If  the  area  of  tlie  normal  section  at  any  point  be  made 
proportional  to  the  tension  at  that  point,  there  will  then  be  a 
constant  tension  per  unit  of  area,  and  the  tendency  to  break 
will  be  the  same  at  all  points.  To  find  the  equation  of  the 
curve  of  equilibrium  under  gravity,  consider  the  equilibrium  of 
an  element  PP'  whose  length  is  c/5,  and  whose  weight  \%  g poods, 
where  00  is  the  section  at  P,  and  p  the  uniform  density.  This 
weight  is  balanced  by  the  difference  of  the  vertical  components 
of  the  tensions  at  /'and  P\  hence 

^(/sin  (p)  =  gpojds, 

d{  T  cos  0)  =  o  ; 

therefore  T  cos  (p  =z  H,  the  tension  at  the  lowest  point,  and 
T  =  H  sec  0.  Again,  if  oo^  be  the  section  at  the  lowest  point, 
then  by  hypothesis  00/ co^  =  T/ H  =  sec  cf),  and  the  first  equation 
becomes 

Hd(sec  (p  sin  (p)  =  gpco^  sec  ((yds, 

or  c  d  id^n  0  =  sec  c/)ds, 

where  c  stands  for  the  constant  H/gpoj^,  the  length  of  string 
(of  section  co^)  whose  weight  is  equal  to  the  tension  at  the 
lowest  point  ;  hence, 

ds  =  c  sec  0^/0,     s/c  =  gd~'0, 
the  intrinsic  equation  of  the  catenary  of  uniform  strength. 

Also     dx  =  ds  cos  0  =  c(^(p,  dy  =  ds  sin  ^  =  c  tan  0  d(p  ; 

hence  .r  =  C(p,  y  =  c  log  sec  0, 

and  thus  the  Cartesian  equation  is 

■y/c  =  log  sec  x/c, 

in  which  the  axis  of  x  is  the  tangent  at  the  lowest  point. 

Prob.  95.  Using  the  same  data  as  in  Art.  3i»  find  the  parameter 
^  and  the  depth  of  the  lowest  point.  (The  equation  x/c  =  gd  s/c 
gives     lo/c  =  gd  i^/c,    which,    by     putting     i^/i'  =  z,    becomes 


50  HYPERBOLIC    FUNCTIONS. 

gd  s  =  fz.  From  the  grapli  it  is  seen  that  z  is  nearly  1.8.  If 
f(z)  =:  gd  2  —  §2,  then,  from  the  tables  of  the  gudermanian  at  the 
end  of  this  chapter, 

/(1.80)  =  1.2432  —  1.2000  =  +  -0432, 

/(1.90)  —  1.2739  —  1.^667  =  +  .0072, 

/(i'95)  —  i-288i  —  1.3000  =  —  .0119, 

whence,  by  interpolation,  2  =  1.91S9  and  c—  78170.  Again, 
yjc  =  logc  sec  x/c  ;  but  xjc  =  10/^  =  1.2793;  ^"d  1-2793  radians 
=  73°  17'  55";  hence^  =  7.8170  X  .54153X2.3026  =  9.7472,  the 
required  depth.) 

Prob.  96.  Find  the  inclination  of  the  terminal  tangent. 

Prob.  97.  Show  that  the  curve  has  two  vertical  asymptotes. 

Prob.  98.  Prove  that  the  law  of  the  tension  T,  and  of  the  section 

a?,   at  a  distance    5,   measured   from   the  lowest   point    along    the 

curve,  is 

T       00  ,    J 

—  =  —  =  cosh  -; 

H        G),  c 

and  show  that  in  the  above  numerical  example  the  terminal  section 
is  3.48  times  the  minimum  section. 

Prob.  99.  Prove  that  the  radius  of  curvature  is  given  by 
o  z=  c  cosh  s/c.  Also  that  the  weight  of  the  arc  s  is  given  by 
/F"  =  H  smh.  s/c,  in  which  s  is  measured  from  the  vertex.. 

Art.  33,    The  Elastic  Catenary. 

An  elastic  string  of  uniform  section  and  densitj-  in  its  natu- 
ral state  is  suspended  from  two  points.  Find  its  equation  of 
equilibrium. 

Let  the  element  da  stretch  into  ds ;  then,  by  Hooke's  law, 
ds  =  d(T{\  -\-  XT),  where  X  is  the  elastic  constant  of  tlie  string; 
hence  the  weight  of  the  stretched  element  ds,  =  jpoodcr,  = 
goa)ds/{i  -{-XT}.     Accordingly,  as  before, 

^r  sin  0)  =  gpoods/{\  -\- XT), 
and  T  cos  (p  =z  H  =  gpcoc, 

hence  <r^(tan  0)  =  ds/{\  -\-  fx  sec  0), 

in  which  //  stands  for  XH,  the  extension  at  the  lowest  point ; 


THE    TRACTORY. 


51 


therefore  ds  =  c{sec''  0  -(-  ;<  sec'  (t>)d(p, 

s/c  =  tan  0  -(-  ^/<(sec  cp  tan  0  +  gd~^  0),     [prob.  20,  p.  37 

which  is  the  intrinsic  eqnation  of  the  curve,  and  reduces  to  that 
of  the  common  catenary  when  /,i  —  o.  The  coordinates  x,  y 
may  be  expressed  in  terms  of  the  single  parameter  0  by  put- 
ting dx  =  ds  cos  0  =  ^(sec  0  4~  /<  sec^  (p)d(p, 

dy  =  ds  sin  0  =  r(sec''  0  +  /<  sec'  0)  sin  0  dcp.     Whence 

x/c  =  gd"'  (p  -\-  ju  tan  0,     j/f  =  sec  0  +  2/'  tan'  0. 

These   equations    are  more  convenient  than  the   result  of 
eliminating  0,  which  is  somewhat  complicated. 

Art.  34.     The  Tractory.* 

To  find  the  equation  of  the  curve  which  possesses  the 
property  that  the  length  of  the  tangent  from  the  point  of  con- 
tact to  the  axis  of  x  is  con- 
stant. 

Let  FT,  P'T'  be  two  con- 
secutive   tangents    such    that 
PT=  P'T'  =  c,  and  let  OT 
=  /;    draw    TS  perpendicular 

to  P'T';   then  U  PP' =  ds,  it      

is    evident    that    ST'    differs      '     ''^  ^        ^' 

from  ds  by  an  infinitesimal  of  a  higher  order.  Let  PT  make 
an  angle  0  with  OA,  the  axis  of  y ;  then  (to  the  first  order  of 
infinitesimals)  PTdcp  =  TS  =  TT'  cos  0;  that  is, 

Cif(f)  =  cos  (pdf,     /  =  r  gd~'0, 

X  =  ^  —  c  sin  0,  =  r(gd~'  0  —  sin  0),    y  =  c  cos  0. 

This  is  a  convenient  single-parameter  form,  which  gives  all 

*  This  curve  is  used  in  Schieie's  anti-friction  pivot  (Minchin's  Statics,  Vol.  i, 
p.  242) ;  and  in  the  theory  of  the  skew  circular  arch,  the  horizontal  projection 
of  the  joints  being  a  tractory.  (See  "Arch,"  Encyclopedia  Britannica.)  The 
equation  0  =  gd  tjc  furnishes  a  convenient  method  of  plotting  the  curve. 


6^ 


HYPERBOLIC    FUNCTIONS. 


values  of  x,  y  3.s  (p  increases  from  o  to  ^,t.  The  value  of  s,  ex- 
pressed in  the  same  form,  is  found  from  the  relation 

ds  =  ST'  =  dt  sin  0  =  ^  tan  <pd(p,     s  ^=  c  log^  sec  (p. 

At  the  point  A,  (p  =  o,  x  =  o,  s  =  o,  /  =  o,  f=c.  The 
Cartesian  equation,  obtained  by  eliminating  (p,  is 

f  =  gd-  (cos-  ^)  -  sin  (cos-  ^^  =  cosh- 1  -  ^,  -^. 

If  u  be  put  for  t/c,  and  be  taken  as  independent  variable, 
(p  =i  gd  //,    x/c  =  u  —  tanh  u,    y/c  =  sech  ti,    s/c  ==  log  cosh  u. 

Prob.  loo.  Given  t  =  2^,  show  that  (p  =  74°  35',  s  =  1.3249^, 
jj'  =  .2658(r,  X  =  1.0360^.     At  what  point  is  /  =  ^  ? 

Prob.  10 1.  Show  that  the  evolute  of  the  tractory  is  tlie  catenary. 
(See  Prob.  92.) 

Prob.  102.  Find  the  radius  of  curvature  of  tlie  tractory  in  terms 
f)^  (p  ;  and  derive  the  intrinsic  equation  of  the  invohite. 

Art.  35.     The  Loxodrome. 

On  the  surface  of  a  sphere  a  curve  starts  from  the  equator 
in  a  given  direction  and  cuts  all  the  meridians   at  the  same 

angle.  To  find  its  equation 
in  latitude-and  longitude  co- 
ordinates : 

Let  the  loxodrome  cross 
two  consecutive  meridians 
AM,  AN\n  the  points/',  Q\ 
let  PR  be  a  parallel  of  lati- 
tude ;  let  OM=x,  MP  =  y, 
MN  ■=  dx,  RQ  =  dy,  all  in  radian  measure  ;  and  let  the  angle 
MOP=  RPQ  =  a;   then 

tan  a  =  RQ/PR,     but     PR  =  JILV  cos  Jl/P* 

hence  dx  tan  a  =  dy  sec  y,  and  x  tan  a  =  gd~'  y,  there  being 
no  integration-constant  since  j/ vanishes  with  x ;  thus  the  re- 
quired equation  is 

J  =  gd  (.1'  tan  (y). 

*  Jones,  Trigonometry  (Ithaca,  1S90),  p.  185. 


COMBINED    FLEXURE    AND    TENSION.  53 

To  find  tlie  length  of  the  arc  0P\  Integrate  the  equation 

ds  =■  dy  CSC  a,     whence  s  ^=^  y  esc  oc. 

To  illustrate  numerically,  suppose  a  siiip  sails  northeast, 
from  a  point  on  the  equator,  until  her  difference  of  longitude  is 
45°,  find  her  latitude  and  distance: 

Here  tan  ex  =.  \,  and  j/  =  gd  x  =  gd  \7t  =  gd  (.7854)  =  .7152 
radians:  s  =  y  V2  =  1.0114  radii.  The  latitude  in  degrees  is 
40.980. 

If  the  ship  set  out  from  latitude  j,,  the  formula  must  be 
modified  as  follows  :  Integrating  the  above  differential  equa- 
tion between  the  limits  (-i',,  j',)  and  {x^,  y^  gives 

{x,  -  ,r,)  tan  or  =  gd  "  >,  -  gd  "  >, ; 

hence  gd"'/^  —  gd~'j',  -{-  {.\\  —  x^)  tan  01,  from  which  the  final 
latitude  can  be  found  when  the  initial  latitude  and  the  differ- 
ence of  longitude  are  given.  The  distance  sailed  is  equal  to 
{y^  —  y,)  CSC  a  radii,  a  radius  being  60  X  i8o/;r  nautical  miles. 
Mercator's  Chart. — In  this  projection  the  meridians  are 
parallel  straight  lines,  and  the  loxodrome  becomes  the  straight 
line  y'  =  x  tan  a,  hence  the  relations  between  the  coordinates  of 
corresponding  points  on  the  plane  and  sphere  are  x'  =  x, 
y'  =  gd~  y.  Thus  the  latitude  y  is  magnified  into  gd  ~  'y,  which 
is  tabulated  under  the  name  of  "  meridional  part  for  latitude 
j"  ;  the  values  of  j/  and  of  7'  being  given  in  minutes.  A  chart 
constructed  accurately  from  the  tables  can  be  used  to  furnish 
graphical  solutions  of  problems  like  the  one  proposed  above. 

Prob.  103.  Find  the  distance  on  a  rhumb  line  between  the  points 
(30°  N,  20°  E)  and  (30°  S,  40"  E). 

Art.  36.    Combined  Flexure  and  Tension. 

A  beam  that  is  built-in  at  one  end  carries  a  load  P  at  the 
other,  and  is  also  subjected  to  a  horizontal  tensile  force  Q  ap- 
plied at  the  same  point;  to  find  the  equation  of  the  curve 
assumed  by  its  neutral  surface:  Let  x,  y  he  any  point  of  the 


64 


HYPERBOLIC    FUNCTIONS. 


elastic  curve,  referred  to  the  free  end  as  origin,  then  the  bend- 
ing moment  for  this  point  is  Qy  —  Px.  Hence,  with  the  usual 
notation  of  the  theory  of  flexure,* 


ax  ax 


P 


Q 
Ef 


'vhich,  on  putting/  —  vix  =  ;/,  audcPj/dx'^  =  (Pu/c/x'',  becomes 


d^u 


dx 


,  =  n'u, 


whence 
that  is, 


u  =-  A  cosh  nx  -\-  B  sinh  nx,         [probs.  28,  30 
y  =.  Dix  -)-  A  cosh  nx  -\-  B  sinh  nx. 


The  arbitrary  constants  A,  B  are  to  be  determined  by  the 
terminal  conditions.  At  the  free  end  a' =  o,  j  =  O ;  hence /i 
must  be  zero,  and 

y  =  inx  -|-  B  sinh  nx, 

—  =.  ni  -\-  hB  cosh  nx  ; 
dx 

but  at  the  fixed  end,  x  =  /,  and  dy/dx  =  o,  hence 

i>  =  —  jji/n  cosh  «/, 


and  accordingly 


y  =  mx 


in  sinh  nx 
n  cosh  ;// 


To  obtain  the  deflection  of  the  loaded  end,  find  the  ordinate 
of  the  fixed  end  by  putting  x  =  I,  giving 

deflection  =  mil—  -tanh;//). 

n  ' 

Prob.  104.  Compute  the  deflection  of  a  cast-iron  beam,  2X2 
inches  section,  and  6  feet  span,  buik-in  at  one  end  and  carrying 
a  load  of  100  pounds  at  the  other  end,  the  beam  being  subjected 
to  a  horizontal  tension  of  8000  pounds.  [In  this  case  /  =  4/3, 
^=15X10',  Q  =  8000,  /*  =  100  ;  hence  n  =  1/50,  w  =  1/80, 
deflection  =  ^17(72  —  50  tanh  1.44)  —  ^^(72  —  4469)  =  -341  inches.] 


^Ier^iman,  Mechanics  of  Materials  ^New  York,  1895),  pp.  70-77,  267-269 


ALTERNATING    CURRENTS.  55 

Prob.  105,   If  the  load  be  uniformly  distributed  over  the  beam, 
say  7U  per  linear  unit,  prove  that  the  differential  equation  is 

EI^,  =  Qv  -  hux\     or      '-A  =  ,i\v  -  nix'), 

2  VI 

and  that  the  solution  is_)'  =  ^  cosh  nx  -\-  B  sinh  ux  \-  tiix'  ^ ^. 

n 

Show  also  how  to  determine  the  arbitrary  constants. 


Art.  37.    Altp:rnating  Currents.* 

In  the  general  problem  treated  the  cable  or  wire  is  regarded 
as  having  resistance,  distributed  capacity,  self-induction,  and 
leakage  ;  although  some  of  these  may  be  zero  in  special 
cases.  Tile  line  will  also  be  considered  to  feed  into  a  receiver 
circuit  of  an}'  description  ;  and  the  general  solution  will  in- 
clude the  particular  cases  in  which  the  receiving  end  is  either 
grounded  or  insulated.  The  electromotive  force  may,  without 
loss  of  generality,  be  taken  as  a  simple  harmonic  function  of 
the  time,  because  any  periodic  function  can  be  expressed  in  a 
Fourier  series  of  simple  harmonics. f  The  E.M.F.  and  the 
current,  which  may  differ  in  phase  by  any  angle,  will  be 
supposed  to  have  given  values  at  the  terminals  of  the  receiver 
circuit  ;  and  the  problem  then  is  to  determine  the  E.M.F. 
and  current  that  must  be  kept  up  at  the  generator  terminals  ; 
and  also  to  express  the  values  of  these  quantities  at  any  inter- 
mediate point,  distant  x  from  the  receiving  end  ;  the  four 
line-constants  being  supposed  known,  viz.: 

r  =  resistance,  in  ohms  per  mile, 

/  =  coefificient  of  self-induction,  in  henrys  per  mile, 

c  =  capacity,  in  farads  per  mile, 

g  =  coefificient  of  leakage,  in  mhos  per  mile.  J 

It  is  shown  in  standard  works§  that  if  any  simple  harmonic 

*  See  references  in  footnote,  Art.  27.  f  Byerly,  Harmonic  Functions. 

t  This  article  follows  the  notation  of  Kennelly's  Treatise  on  the  Application 
of  Hyperbolic  Functions  to  Electrical  Engineering  Problems,  p.  70. 

§  Thomson  and  Tait,  Natural  Philosophy,  Vol.  I.  p.  40;  Raleigh,  Theo'y  of 
Sound,  Vol.  I.  p.  20;  Bedell  and  Crehore,  Alternating  Currents,  p.  214. 


56  HYPERBOLIC    FUNCTIONS. 

function  a  sin  (&»/  -(-  S)  be  represented  by  a  vector  of  length 
a  and  angle  d,  then  two  simple  harmonics  of  the  same  period 
2n/cj,  but  having  different  values  of  the  phase-angle  0,  can  be 
combined  by  adding  their  representative  vectors.  Now  the 
E.M.F.  and  the  current  at  any  point  of  the  circuit,  distant  x 
from  the  receiving  end,  are  of  the  form 

e  =  e^  sin  {cot  -{-  H),     i  =  /,  sin  {oot  -)-  B'),  (64) 

in  which  the  maximum  values  <',,  /,,  and  the  phase-angles  B,  B', 
are  all  functions  of  x.  These  simple  harmonics  will  be  repre- 
sented by  the  vectors  eJB,  ijd' ;  whose  numerical  measures 
are  the  complexes  r,  (cos  B  -f-y' sin  ^)*,  /,  (cos  B'  -\- j  sin  B'), 
which  will  be  denoted  hye,i.  The  relations  between /and  i 
may  be  obtained  from  the  ordinary  equations  f 

di  de      de  di 

for,  since  de/dt  =  ooe^  cos  (w/  -\-  6)  =  wCi  sin  (co/  +  ^  +  §7r),  then 
de/dt  will  be  represented  by  the  vector  oiei/d-\-  ^ir]  and  di/'dx 
by  the  sum  of  the  two  vectors  gex/d,  Cijie^/d  -\-\ir\  whose 
numerical  measures  are  the  complexes  ge,  juce;  and  similarly 
for  de/dx  in  the  second  equation  ;  thus  the  relations  between 
the  complexes  e,  i  are 

^  =  (^  +  icoOe,      ;£.  =  (''  +  i"0*'-  (66)t 

*  In  electrical  theory  the  symbol  j  is  used,  instead  of  /,  for  '♦^  —  i. 

t  Bedell  and  Crehore,  Alternating  Currents,  p.  181.  The  sign  of  dx  is 
changed,  because  .v  is  measured  from  the  reccivmg  end.  The  coefficient  of 
leakage,  g,  is  usually  taken  zero,  but  is  here  retained  for  generality  and  sym- 
metry. 

I  These  relations  have  the  advantage  of  not  involving  the  time.  Steinmetz 
derives  them  from  first  principles  without  using  the  variable  /.  For  instance, 
he  regards  r  -\-  joil  as  a  generalized  resistance-coeflicicnt,  which,  when  applied 
to  i,  gives  an  E.M.F.,  part  of  which  is  in  phase  with  /,  and  part  in  quadrature 
with  /.  Kennelly  calls  r  +  j^^l  the  conductor  impedance;  and  g  -\-  juc  the 
dielectric  admittance;  the  reciprocal  of  which  is  the  dielectric  impedance. 


ALTERNATING   CURRENTS 


57 


Differentiating  and  substituting  give 
^2=  (.''  +  i^Oig  -^jc^c)e, 


dH 


dx 


:.  =  (r  +  J^Ois  +  j^<^)i' 


(67) 


and   thus  e,   I  are   similar  functions  of  x,  to  be  distinguished 
only  by  their  terminal  values. 

It  is  now  convenient   to   define   two  constants  a,  So  by  the 
equations  * 

«2  ^(^r  +  ju^l)  (g  +  jo^c) ,      z,  =  a/{g  +  ji^c)  ;  (68) 

and  the  differential  equations  may  then  be  written 


C?2g 


dH 


-1—.,  —  a-e,       -r-,  =  a-t, 
dx-  dx- 


(69) 


the  solutions  of  which  are  f 

e  =^  A  cosh  ax  +  ^  sinh  ax,      i  =  A'  cosh  ax  -\-  B'  sinh  ax, 

wherein  only  two  of  the  four  constants  are  arbitrary;  for 
substituting  in  either  of  the  equations  (66),  and  equating 
coefficients,  give 

{g-^io:c)A=aB',       {g-^  jooc)B  =  aA', 

whence  B' =  A/zo,     A' =  B/z^. 

Next  let  the  assigned  terminal  values  of  e,  t,  at  the  re- 
ceiver be  denoted  hy  E,  /;  then  putting  x  =  O  gives  E=  A, 
I  =  A',  whence  B  =  zj,  B'  =  E/zq]  and  thus  the  general  so- 
lution is 


e  =  E  cosh  ax  +  ZqI  sinh  ax, 


i  =  I  cosh  ax  H E  sinh  ax, 

2o 


(70) 


*  Professor  Kennelly  calls  a  the  attenuation-constant,  and    So  the    surge- 
impedance  of  the  line. 

t  See  Art.  14,  Probs.  28-30;  and  Art.  27,  foot-note. 


58  Hyperbolic  functions. 

If  desired,  these  expressions  could  be  thrown  into  the  ordi- 
nary com})Iex  form  X -\-  jY,  X'  -\-jV',  by  putting  for  the  let- 
ters their  complex  values,  and  applying  the  addition-theorems 
for  the  hyperbolic  sine  and  cosine.  The  quantities  X,  Y,  X', 
Y'  would  then  be  expressed  as  functions  of  x ;  and  the  repre 
sentative  vectors  of  e,  i,  would  be  i\/0,  z,  /8\  where  ^/  =  A'^-[~  ^S 
/;  =  X"  +  Y'\  tan  0  =  Y/X,  tan"^  =~Y'/X. 

For  purposes  of  numerical  computation,  however,  the  for- 
mulas (70)  are  the  most  convenient,  when  either  a  chart,*  or  a 
table, f  of  cosh  //,  sinh  u,  is  available,  for  complex  values  of  ?/. 

Prob.  106. J  Given  the  four  line-constants:  r  =  2  ohms  per 
mile,  f  =  20  millihenrys  per  mile,  c  =  1/2  microfarad  per  mile, 
g  =  o;  and  given  co,  the  angular  velocity  of  E.M.F.  to  be  2000 
radians  per  second;  then 

0)1  =  40  ohms,  conductor  reactance  per  mile; 
r  +  /co/  =  2  +  40/  ohms,  conductor  impedance  per  mile; 

uc  =  .GDI  mho,  dielectric  susceptance  per  mile; 
g  +  juc  =  .001;  mho,  dielectric  admittance  per  mile; 
(g  -|-  /aj'~)~*  =  —  1000/  ohms,  dielectric  impedance  per  mile; 

a-  =  (r+  j'cjoI)  (g  +  /wc)  =.04  +.002/,  which  is  the  measure 

of  .04005   177°  8';  therefore 
a  =  measure  of  .2001  88°  34'  =  .0050  +  .2000;,  an  ab- 
stract coeflficient  per  mile, of  dimensions  [length]"  S 
z^  =  a/{g  +  /coc)  =  200  —  5/  ohms. 

Next  let  the  assigned  tenninal  conditions  at  the  receiver  be ; 
7  =  0  (line  insulated);  and  E  =  1000  volts,  whose  phase  may  be 
taken  as  the  standard  (or  zero)  phase ;  then  at  any  distance  x, 
by  (70), 

E 

e  =  E  cosh  ax,     ^  =  ~  sinh  ax, 

in  which  ax  is  an  abstract  complex. 

Suppose  it  is  required  to  find  the  E.M.F.  and  current  that 
must  be  kept  up  at  a  generator  100  miles  away;  then 

*  Art.  30,  foot-note.  t  See  Table  II. 

X  The  data  for  this  example  are  taken  from  Kennelly's  articif'  (1.  c. 
p.  38). 


ALTERNATING    CURRENTS.  69 

e  —  looo  cosh  (.5  -f-  20/),     I  =  200(40  —  jY^  sinh  (.5  -\-  20J), 
but,  by  page  44,  cosh  (.5  +  2oj)  =  cosh  (.5  +  2oy  —  GttJ) 

=  cosh  (.5  +  1. 15/)  =  .4600  +  .4750/' 

obtained  from  Table  II,  by  interpolation  between  cosh  (.5  +  i.y) 
and  cosh  (.5  -(-  1.27);   hence 

e  —  460  +  475/'=  -^.(cos  6^4- /sin  /^), 

where    log  tan    ^  =  log  475  -  log  460  =  .0139,   (^  =  45°    55',     and 
e^  —  460  sec  0  =  661.2  volts,  the  required  E.M.F. 

Similarly  sinh  (.5  +  207)  =  sinh  (.5  +  i-iSy)  =  .2126+  1.0280/, 
and  hence 

^'"^  7a°£(4o  +  /)(.2i26  +  1.028/)  =  -7— (1495  +  8266/) 

lOOI  lOOI 

=  /,(cos  0'  -{- J  sin  6'), 
where  log  tan  6'  =  10.7427,  6'  =  79°  45',  /,  =  1495  sec  ^'/i6oi  — 
5.25  amperes,  the  phase  and  magnitude  of  required  current. 

Next  let  it  be  required  to  find  ^  at  .v  =  8;   then 

^=  1000  cosh  (.04  -(-  i.6y)  =  1000/ sinh  (.04+  -os/)* 

by    subtracting   ^tt/,   and    applying     page   44.        Interpolation  be- 
tween sinh  (0  +  0;)  and  sinh  (o  -f-  .1/)  gives 

sinh  (o  -f-  -03/)  =  00000  +  .02995/. 

Similarly  sinh  (.1  -f  -03/)  =  .10004-)-  ■03oo47' 

Interpolation  between  the  last  two  gives 

sinh  (.04  -[-  .03/)  =  .04002  -\-  .02999/ 

Hence  r  =y(^o. 02  +29.99/)=  —  29.994-40.02/ =^, (cos  B-{-j?>\n  H), 
where 

log  tan  6  =  .12530,  ^  =  126°  51',^,  =  —  29.99  s^c  126°  51'  =  50.01 
volts. 

Again,  let  it  be  retpiired  to  find  e  Vit  x  =  16;  here 

e  —  lOoo  cosh  (.08  +  3.2/)  =  —  1000  cosh  (.08  -\-  .o6j), 

but  cosh  (o  -|-  .o6y)  =  .997°  +  o/,  cosh  (. i  -j-  .06/)  =  1.0020  -|-  .006/; 

hence  cosh  (.08 -)- .06/)  =  1.0010 -{-.0048/, 

and  ^=  —  iooi-|-4.8/=  c'll^cos  ^-f-ysin  ^), 

where  ^  —  180°  17',  e^  =  looi  volts.     Thus   at   a  distance  of  about 
16  miles  the  E.M.P'.  is  the  same  as   at  the  receiver,  but  in  opposite 


60  HYPERBOLIC    FUNCTIONS. 

phase.  Since  c  is  proporlional  to  cosh  (.005  -|-  •2j)x,  the  value  of 
X  for  which  the  phase  is  exactly  180°  is  tt/.z  —  15.7.  Similarly 
the  phase  of  the  E.AI.F.  at  x  =  7.85  is  90°.  There  is  agreement 
in  phase  at  any  two  points  whose  distance  apart  is  31.4  miles. 

In  conclusion  take  the  more  general  terminal  conditions  in 
which  the  line  feeds  into  a  receiver  circuit,  and  suppose  the  current 
is  to  be  kept  at  50  amperes,  in  a  phase  40°  in  advance  of  the  elec- 
tromotive force;  tiien  /  —  5o(cos  40°  +/  sin  40°)  =  38.30  +  32-i4/> 
and  substituting  the  constants  in  (70)  gives 

e  =z  icoo  cosh  (.005  -\-  .2j)x  -\-  (7821  -j-  6216J)  sinh  (.005  -f  -V)-^ 
—  4604  4757  -4748+93667=  -4288+984iy=r,(cos  ^-fy  sin  ^), 

where  ^=  113°  ZZ'y^x  —  '°73°  volts,  the  E.M.F.  at  sending  end, 
This  is  17  times  what  was  required  when  the  other  end  was  insulated. 
Prob.   107.    If  /  =  o,    g  =  o,    7=0;    then   a={i-{-j)n,   Zo  = 
(i  +y)Mi,  where  n^  =  core  '2,  w,-  =  r/2c«jc;  and  the  solution  is 


^1  —  T7=£  t^cosh  2nx  +  cos  2nx,     tan  6  —  tan  nx  tanh  nx, 
^  2 


ii  =  — E  I  cosh  2nx  —  cos  2nx,    tan  6'  =  tan  nx  coth  nx. 

Prob.  108.  If  self-induction  and  capacity  be  zero,  and  the  re- 
ceiving end  be  insulated,  show  that  the  graph  of  the  electromotive 
force  is  a  catenar}^  if  g  ?^  o,  a  line  if  g  =  o. 

Prob.  log.  Neglecting  leakage  and  capacity,  prove  that  the 
solution  of  equations  (66)  isi  —  I,e  —  E-\-(r-\-  juljix. 

Prob.  no.  If  a;  be  measured  from  the  sending  end,  show  how 
equations  (65),  (66)  are  to  be  modified;  and  prove  that 

e  =  Eo  cosh  ax  —  zJo  sinh  ax,     I  —  h  cosh  ax En  sinh  ax, 

_    _  •^0 

where  £c  lo  refer  to  the  sending  end. 

Art.  38.     Miscellaneous  Applications. 

1.  The  length  of  the  arc  of  the  logaritlimic  curve  y  =  <7*  is 
S  =  M{cosh  //-(-logtanli  |?/),  in  which  Al=  i/log  a,  sinh  u  —  y/M. 

2.  The  length  of  arc  of  the  spiral  of  Archimedes  r  =^  a^  xs, 
s  =  i(7(sinh  2ti  -j-  2//),  where  sinh  21  =  6>. 

3.  In  the  hyperbola  x^ /a"  —  y' /b'  =  i  the  radius  of  curva- 
ture is  p  =  {a'  sinh'  u -\- h'  cosh'  iif/ab;  in  which  u  is  the 
measure  of  the  sector  AOP,  i.e.  cosh  u  =  x/a,  sinh  //  z=y/b. 

4.  In  an  oblate  spheroid,  the  superficial  area  of  the  zone 


MISCELLANEOUS    APPLICATIONS.  61 

between  the  equator  and  a  parallel  plane  at  a  distance  j  is 
5  =  iTT/iXsinh  2u  -|-  2u)/2e,  wherein  b  is  the  axial  radius,  e  eccen- 
tricity, sinh  u  =  ey/p,  and  /  parameter  of  generating  ellipse. 

5.  The  length  of  the  arc  of  the  parabola  jj/'  =  2px,  measured 
from  the  vertex  of  the  curve,  is  /  =  5/'(sinh  2u-\-2u\  in  which 
sinh  u  ^=y/p  =  tan  0,  where  0  is  the  inclination  of  the  termi- 
nal tangent  to  the  initial  one. 

6.  The  centre  of  gravity  of  this  arc  is  given  b^' 

2,lx  =r/'(^cosh'  u  —  i),     64/]'  =:  p"  (sinh  411  —  4//) ; 

and  the  surface  of  a  paraboloid  of  revolution  is  5  =  2ti yl. 

7.  The  moment  of  inertia  of  the  same  arc  about  its  ter. 
minal  ordinate  is  /=  ;/[,r/(^  —  2^)  -f  ^'^/^W],  where  /<  is 
the  mass  of  unit  length,  and 

JSl  =i  II  —  '^  sinh  2n  —  ^  sinh  4?{-\~  y^^  sinh  6//. 

8.  The  centre  of  gravit)'  of  the  arc  of  a  catenary  measured 
from  the  lowest  point  is  given  by 

4/y=  ^'(sinli  2?/  -\-  211),  !x  =  c^{ii  sinh  ii  —  cosh  ?/  -f~  i)j    • 

in  which  ?(  =x/c;  and  the  moment  of  inertia  of  this  arc  about 
its  terminal  abscissa  is 

/  =  J^c'Xj^iy  sinh  3//  -\-  f  sinh  ?/  —  7/  cosh  ?i). 

9.  Applications  to  the  vibrations  of  bars  are  given  in  Ray- 
leigh,  Theory  of  Sound,  Vol.  I,  art,  170:  to  the  torsion  of 
prisms  in  Love,  Elasticity,  pp.  166-74;  to  the  flow  of  heat 
and  electricity  in  Byerly,  Fourier  Series,  pp.  75-81  ;  to  wave 
motion  in  fluids  in  Rayleigh,  Vol.  I,  Appendix,  p.  477,  and  in 
Bassett,  Hydrodynamics,  arts.  120,  384;  to  the  theory  of 
potential  in  Byerly  p.  135,  and  in  Maxwell,  Electricity,  arts. 
172-4;  to  Non-Euclidian  geometry  and  many  other  subjects 
in  Giinther,  Hyperbelfunktionen,  Chaps.  V  and  VI.  Several 
numerical  examples  are  worked  out  in  Laisant,  Essai  sur  les 
fonctions  hyperboliques. 


b'-J  HYPKRHOLIC    FUNCTKjNS. 

Art.  39.     Explanation  of  Tables. 

In  Table  I  the  numerical  values  of  the  hyperbolic  functions 
sinh  II,  cosh  n,  tanh  u  are  tabulated  for  values  of  u  increasing 
from  o  to  4  at  intervals  of  .02.  When  ti  exceeds  4,  Table  IV 
may  be  used. 

Table  II  gives  hyperbolic  functions  of  complex  arguments, 
in  which 

cosh  {x  ±  iy)  =  «  ±  ib,     sinh  [x  ±  iy)  =  c  ±_  id, 

and  the  values  of  a,  b,  c,  d  are  tabulated  for  values  of  x 
and  o[  y  ranging  separately  from  o  to  1.5  at  intervals  of  .1. 
When  interpolation  is  necessary  it  may  be  performed  in  three 
stages.  For  example,  to  find  cosh  (.82 -[-  1-340  •  Fh'st  find 
cosh  (.82 -j-  1.3/),  by  kecpingj'at  1.3  and  interpolating  between 
the  entries  under  a"  =  .8  and.r  =  .9  ;  next  find  cosh  (.82  -f  1. 4/), 
by  keeping  ^^^  at  1.4  and  interpolating  between  the  entries  under 
;ir  =  .8  and  x  ^  .9,  as  before;  then  by  interpolation  between 
cosh  (.82  -(-  1.3/)  and  cosh  (.82  -|-  i-40  ^""^^  cosh(  .82  -f  1-340' 
in  which  x  is  kept  at  .82.  The  table  is  available  for  all  values 
of  _^,  however  great,  b\-  means  of  the  formulas  on  page  44: 

sinh  (.r  -]-  2/'T  )  =  sinh  a',     cosh  {x  A^  2/t)  =  cosh  x,  etc. 

It  does  not  apply  when  x  is  greater  than  1.5,  but  this  case  sel- 
dom occurs  in  practice.  This  tabic  can  also  be  used  as  a  com 
plex  table  of  circular  functions,  for 

cos  {y  ■±_  ix)  =  cr  =p  //;,     sin  {y  ±  ix)  ^  d  ±_ic  ', 

and,  moreover,  the  cxponenticd  function  is  given  by 

exp  {±x  ±  ty)  z=a±c  ±  ;(/;  ±  d), 

in  which  the  signs  of  c  ami  ^/are  to  be  taken  the  same  as  the 
sign  of  X,  and  the  sign  of  i  on  the  right  i^;  to  be  the  product  of 
the  signs  of  x  and  of  i  on  the  left.     (See  A[)pendix,  C.) 

Table  III  gives  the  values  of  v—  gd  n,  and  of  the  guder- 
manian  angle  6=  180°  I'/n,  as  ti  changes  from  o  to  I    at  inter- 


EXPLANATION    OF    TABLES.  ti3 

vals  of  .02,  from    i   to  2  at  intervals  of  .05,  and  froiri  2  to  4  at 
intervals  of  .1. 

In  Table  IV  are  given  the  values  of  gd  u,  log  sinh  ?/,  log 
cosh  u,  as  u  increases  from  4  to  6  at  intervals  of  .1,  from  6  to 
7  at  intervals  of  .2,  and  from  7  to  9  at  intervals  of  .5. 

In  the  rare  cases  in  which  more  extensive  tables  are  neces- 
sary, reference  may  be  made  to  the  tables*  of  Gudermann, 
Glaisher,  and  Geipel  and  Kilgour.  In  the  first  the  Guderman- 
ian  angle  (written  k)  is  taken  as  the  independent  variable,  and 
increases  from  o  to  100  grades  at  itUervals  of  .01,  the  corre- 
sponding value  of  u  (written  Lk)  being  tabulated.  In  the  usual 
case,  in  which  the  table  is  entered  with  the  value  of  //,  it  gives 
by  interpolation  the  value  of  the  gudermanian  arigle,  whose 
circular  functions  would  then  give  the  hyperbolic  functions 
of  u.  When  it  is  large,  this  angle  is  so  nearl\'  right  that  inter- 
polation is  not  reliable.  To  remedy  this  inconvenience  Gu- 
dermann's  second  table  gives  directly  log  sinli //,  log  cosh;/, 
log  tanh  //,  to  nine  figures,  for  values  of?/  var}'ing  by  .OOI  from 
2  to  5,  and  by  .01  from  5  to  12. 

Glaisher  has  tabulated  the  values  of  r*  and  c'",  to  nine  sig- 
nificant figures,  as  x  varies  by  .001  from  o  to  .[,  by  .01  from  O 
to  2,  by  .1  from  o  to  10,  and  by  i  from  o  to  500.  From  these 
the  values  of  cosh  x,  sinh  x  are  easily  obtained. 

Geipel  and  Kilgour's  handbook  gives  the  values  of  coshjt, 
sinh  X,  to  seven  figures,  as  x  varies  by  .01  from  o  to  4. 

There  are  also  extensive  tables  by  Forti,  Gronau,  Vassal, 
Callet,  and  Hoiiel ;  and  there  are  four-place  tables  in  Byerly's 
Fourier  Series,  and  in  Wheeler's  Trigonometry,  (See  Ap- 
pendix, C.) 

In  the  following  tables  a  dash  over  a  final  digit  indicates 
that  the  number  has  been  increased, 

*Gudermann  in  Crelle's  Journal,  vols.  6-9,  1831-2  (published  separately 
under  ihe  title  Theorie  der  hyperbolischen  Functionen,  Berlin,  1S33).  Glaisher 
in  Cambridge  Phil.  Trans.,  vol.  13,  1881.  Geipel  and  Kilgour's  Electrical  Hand- 
book. 


64 


HYPERBOLIC    FUNCTIONS. 


Table  I. —  Hyperbolic  Functions. 


u. 

sinh  u. 

cosh  u. 

tanh  ». 

u. 

sinh  «. 

cosh  u. 

tanh  u. 

.00 

.0000 

1.0000 

.0000 

1.00 

1.1752 

1.5431 

.7616 

02 

0200 

1.0002 

0200 

1.02 

1.20  3 

1  5669 

7699 

04 

0400 

1.00C8 

0400 

1.04 

1  2379 

1.5913 

7779 

06 

0600 

1.0018 

0599 

1.06 

1  2700 

1  6164 

7857 

08 

0801 

1.0032 

0798 

1.08 

1.302) 

1.6421 

7932 

.10 

.1002 

1.0050 

.0997 

1.10 

1  3356 

1  6685 

.8005 

12 

1203 

1  0072 

1194 

1.12 

1  3693 

1.6956 

8076 

14 

1405 

1 .0098 

1391 

1.14 

1  40:!5 

1.7233 

8144 

16 

1607 

1.01  ••8 

1586 

1.16 

1.4382 

1.7517 

8210 

18 

1810 

1.0102 

1781 

1.18 

1.4735 

1.78U8 

8275 

.20 

.2013 

1.0201 

.1974 

120 

1  5095 

1  8107 

.8337 

22 

2218 

1.0243 

2165 

1.22 

1.54.0 

1.8412 

8397 

24 

2423 

1.0289 

2355 

1.24 

1.5831 

1.8720 

8455 

26 

2029 

1.0340 

2543 

1.26 

1.6209 

1.9045 

8511 

28 

2837 

1.0395 

2729 

1.28 

1.6593 

1.9373 

856.5 

.30 

.3045 

1  0453 

.2913 

1.30 

1.6984 

1.9709 

.8617 

32 

3255 

1  0516 

3095 

1.32 

1.7381 

2.0053 

8668 

34 

3466 

1.0584 

3275 

1  34 

1.77^6 

2  0404 

8717 

m 

3618 

1.0655 

3452 

1.36 

1.8198 

2.0764 

8764 

38 

3892 

1.0731 

3627 

1.38 

1.8617 

2.1132 

8810 

.40 

.4108 

1.0811 

.3799 

1.40 

1.9043 

2  1509 

.8854 

42 

43-.J5 

1.0895 

3969 

1.42 

1.9477 

2  1894 

8896 

44 

4543 

1.0984 

4136 

1.44 

19919 

2.228S 

8937 

46- 

47(54 

1.1077 

4301 

1.46 

2.0369 

2.2691 

8977 

48 

'4986 

1.1174 

4462 

1.48 

2.0827 

2.3103 

9015 

.50 

.5211 

1.1276 

.4621 

1.50 

2.1293 

2.3524 

.9051 

52 

5)38 

1.1383 

4777 

1  52 

2  1768 

2.3955 

9087 

54 

5666 

1  1494 

4930 

1.54 

2.2251 

2.4395 

9121 

56 

5897 

1.1609 

5080 

1.56 

2.2743 

2.4845 

9154 

58 

6131 

1.1730 

5227 

1.58 

2.3245 

2.5305 

9186 

.60 

.6367 

1  1855 

.5370 

1.60 

2.3756 

2.5775 

9217 

62 

6605 

1.1984 

5511 

1.62 

2.4276 

2  6255 

9246 

64 

6846 

1  2119 

5649 

1.64 

2.4806 

2.6746 

9275 

66 

7090 

1.2258 

5784 

1.66 

2  5346 

2.7247 

9302 

68 

7336 

1.2402 

5915 

1.68 

2.5896 

2.7760 

9329 

.70 

.7586 

1.2552 

.6044 

1.70 

2.6456 

2.8283 

.9354 

72 

7888 

1.2706 

6169 

1.72 

2.7027 

2.8818 

9379 

74 

8094 

1  2S65 

6-.' 9 1 

1  74 

2.7609 

2.9364 

9402 

76 

8353 

1.3030 

6411 

1.76 

2.8202 

2  9922 

9425 

78 

8615 

1.3199 

6527 

1.78 

2.8806 

3.0492 

9447 

.80 

.8881 

1.3374 

.6040 

1.80 

2.9422 

3  1075 

.9468 

82 

9150 

1.3555 

6751 

1.82 

3.0049 

3.1669 

9488 

84 

9423 

1  3740 

685S 

1.84 

3.0689 

3  2277 

9508 

86 

9700 

1  3932 

6963 

1.86 

3.1340 

3  2897 

95':7 

88 

9981 

1.4128 

7004 

1.88 

3.2005 

3  3530 

9545 

.90 

1.0265 

1.4331 

.716:1 

1.90 

3.2682 

3.4177 

.9562 

92 

1 . 0554 

1.4539 

7259 

1.92 

3.3372 

3  4S38 

9579 

94 

1  0S47 

1  4753 

7352 

1  94 

3.4075 

3.5512 

9595 

96 

1.1144 

1.4973 

7443 

1.96 

3  4792 

3.6201 

961  i 

98 

1  1446 

1.5199 

7531 

1.98 

3.5523 

3  6904 

9626 

TABLES, 


65 


Table  I.     Hyperbolic  Functions. 


». 

sinh  u. 

cosh  «. 

tanh  u. 

u. 

sinl)  u. 

cosh  u. 

tanh  u. 

2.00 

3.6269 

3.7622 

.9640 

300 

10.0179 

100677 

.99505 

2.02 

8.7028 

3.88.-)5 

9654 

3.02 

10.2212 

10.2700 

99524 

2.04 

3.78U8 

89108 

9667 

8.04 

10.4287 

10.4765 

99.543 

2.06 

3.8598 

8.9867 

9680 

3  06 

10.6408 

10  6872 

99561 

2.08 

8  9898 

4.0647 

9698 

3.08 

10.8562 

10.9022 

99578 

2.10 

4.0219 

4  1443 

.9705 

3.10 

11.0765 

11.1215 

.99.594 

2.12 

4.1056 

4.2256 

97  It; 

3  12 

11.8011 

11.3453 

99610 

2  14 

4  1909 

4.3085 

9727 

3.14 

11  5803 

11  57  £,6 

99626 

2.16 

4  2779 

4.3982 

9787 

3.16 

11  7641 

11.8065 

99640 

2.18 

4.3666 

4.4797 

9748 

3.18 

12.0026 

12.0442 

99654 

2.20 

4  4571 

4.5679 

.  9757 

8.20 

12.2459 

12  2866 

.99668 

2.22 

4.5494 

4.6510 
4.7499 

9767 

8.22 

12.4941 

12.5340 

99681 

2.24 

4.6484 

9776 

3.24 

12.7473 

: 2  7^  64 

99693 

2.26 

4  7894 

4.8487 

978.5 

3  26 

18  0056 

18.0440 

99705 

2  28 

4.8872 

4  9895 

9793 

3.28 

13.2691 

13.3067 

99717 

2.80 

4.9370 

5.0.^.72 

.9801 

3.80 

13.5379 

13  5748 

.99728 

2  82 

5  0887 

5.1370 

9809 

3  82 

18.8121 

13  H483 

99788 

2.84 

5.1425 

5  2888 

9816 

3.84 

14.0918 

14.1278 

99749 

2.86 

5  2488 

5.3427 

9828 

3.86 

14.8772 

i4.4i::o 

99758 

2.38 

5.3562 

5  4487 

9880 

3.88 

14.6684 

14.7024 

99768 

2.40 

5.4662 

5.5569 

.9887 

3.40 

14  96.54 

14.9987 

.99777 

2.42 

5.5785 

5.6674 

9843 

3.42 

15.2684 

15  80U 

99786 

2.44 

5.6929 

5.7801 

9849 

3.44 

15.  .5774 

15  6095 

99794 

2.46 

5  8097 

5  8951 

9855 

3  46 

15.8928 

15.9242 

99802 

2.48 

5.9288 

6.0125 

9861 

3.48 

16.2144 

16.2453 

99810 

2.50 

6  0502 

6  1323 

.9866 

3  50 

16.5426 

16.  .5728 

.99817 

2  52 

6.1741 

6.2545 

9871 

3  52 

16.8774 

1 6  9070 

99824 

2.54 

6.8004 

6.8793 

9876 

3.54 

17.2190 

17,2480 

99831 

2.56 

6.4293 

6.5066 

9881 

3.56 

17.5674 

17.59,58 

99838 

2.58 

6.5607 

6.6364 

9886 

3.58 

17.9228 

17.9507 

99844 

2.60 

6  6947 

6.7690 

.9890 

3.60 

18.2854 

18.8128 

.99850 

2.62 

6.8815 

6.9043 

989.5 

3.62 

18.6554 

18.6822 

99856 

2.64 

6  9709 

7.04  3 

9899. 

3.64 

19.0328 

19.0590 

99862 

2.66 

7.1132 

7  1882 

9903 

3.66 

19.4178 

19.4435 

99867 

2.68 

7.2583 

7.3268 

9906 

3.68 

19.8106 

19.83-,8 

99872 

2.70 

7.4068 

7.4735 

.9910 

3.70 

20.2113 

20.2360 

.99877 

2.72 

7.5572 

7.6281 

9914 

3.72 

20  6201 

20.6443 

99882 

2.74 

7.7112 

7.7758 

9917 

3.74 

21.0871 

21.0609 

99887 

2.76 

7.8683 

7.9816 

9920 

3.76 

21.4626 

21.4859 

99891 

2  78 

8  0285 

8.0905 

9923 

3.78 

21.8966 

21.9194 

99896 

2.80 

8.1919 

8.2527 

.9926 

3.80 

22.3394 

22  8618 

.99900 

2  82 

8  3.">86 

8.4182 

9929 

3.82 

23.7911 

22  8181 

99904 

2.84 

8.5287 

8.5871 

9932 

3  84 

23.2.520 

28.2735 

99907 

2.86 

8  7021 

8.7.594 

9985 

3.86 

28.7221 

28.7432 

99911 

2  88 

8  8791 

8.9352 

99:!7 

3  88 

24.2018 

24.2224 

99915 

2  90 

9  0596 

9.1146 

.9940 

3  90 

24  6911 

24.7113 

.99918 

2.92 

9.2487 

9.2976 

9942 

8.92 

25  1908 

25.2101 

99921 

2.94 

9.481.5 

9  4844 

9944 

3.94 

25.6996 

25.7190 

99924 

2  96 

9  6281 

9  6749 

9947 

3.96 

26  2191 

26.2382 

99927 

2.98 

9  8185 

9.8693 

9949 

3.98 

26.7492 

26.7679 

99930 

66 


HYPERBOLIC    FUNCTIONS. 

Table   II.     V^alues  of  cosh  (x  +  ij')  and  sinh  (x  -f  I'v). 


X   = 

o 

X  =  .\ 

y 

a 

b 

c 

d 

a 

b 

c 

d 

0 

1.0000 

0000 

0000 

.0000 

1.0050 

.00000 

.10017 

.0000 

.1 

().'J9.-.0 

" 

0998 

1 . 0000 

01000 

09967 

1003 

.3 

0.9801 

" 

19S7 

0.9850 

0199(1 

0;)817 

1997 

.3 

0.9.j5:j 

" 

2955 

0.9G01 

02960 

09570 

2970 

.4 

.9211 

.' 

.3  m 

.9257 

.03901 

.09226 

.39)4 

.5 

8776 

" 

4?  94 

8820 

04802 

08791 

4S1H 

.6 

82.-)3 

" 

5046 

82J5 

05(;50 

08-J67 

5675 

.7 

7648 

" 

6442 

7687 

06453 

07661 

U74 

.8 

.6967 

.< 

.7174 

.7002 

.07186 

.06979 

.7200 

.9 

6216 

" 

7S33 

0247 

07847 

06227 

7872 

1.0 

5403 

" 

8415 

5430 

08429 

05412 

8457 

1.1 

4536 

" 

8912 

4559 

08927 

04544 

8957 

1.2 

.3624 

" 

.9330 

.3642 

.09336 

.03630 

0  9367 

1.3 

2675 

" 

9()36 

2688 

09652 

02().so 

0.9684 

1.4 

1700 

" 

9851 

1708 

09871 

01703 

0 . 9904 

1.5 

0707 

' ' 

99  7o 

0711 

09992 

00709 

1.0025 

\^ 

0000 

" 

1.0000 

0000 

10017 

00000 

1.0050 

y 

X   = 

•  4 

X   = 

•5 

a 

b 

c 

d 

a 

b 

c 

,/ 

0 

1  0811 

.0000 

.4108 

.0000 

1.1276 

.0000 

.5211 

.0000 

.1 

1.0 :5() 

0410 

4087 

1079 

1.1220 

0520 

518.5 

1126 

.2 

1.0595 

0816 

4026 

2148 

1.1051 

1025 

5107 

2240 

.3 

1.0328 

1214 

3924 

3195 

1.0773 

1540 

4978 

3332 

.4 

.9957 

.1600 

.3783 

.4210 

1.0386 

.2029 

.4800 

.4391 

.5 

9487 

1969 

361)5 

5183 

0.9896 

2498 

4573 

5406 

.6 

8922 

2319 

3390 

6104 

0.9306 

2942 

4301 

6367 

.7 

8268 

2646 

3142 

6964 

0.8624 

3357 

3986 

7264 

.8 

.7532 

.2947 

.2862 

.7755 

.7856 

.3738 

.3631" 

0.8089 

.9 

6720 

3218 

2553 

8468 

7009 

4082 

3239 

0.S833 

1.0 

5841 

3456 

2219 

9097 

6093 

4385 

2815 

0.9489 

1.1 

4904 

3661 

1863 

9(i3.5 

5115 

4644 

23(i4 

1.0050 

1.2 

.3917 

.3829 

.1488 

1.0076 

.4086 

.4857 

.18S8 

1.0510 

1.3 

2892 

3958 

1099 

1.0417 

3016 

5021 

1394 

1.0865 

1.4 

1838 

4048 

0698 

1.0653 

1917 

5135 

0886 

1.1163 

1.5 

0765 

4097 

0291 

1.0784 

0798 

5198 

0369 

1.1248 

\Tl 

0000 

4108 

0000 

l.OSli 

0000 

5211 

0000 

1.1276 

TABLES. 

Table  II.     Values  of  cosh  {x  -f  ijy)  and  sinh  (x  -|-  jy). 


67 


X   = 

.2. 

X    = 

•3 

a 

^ 

C 

d 

a 

b 

c 

d 

y 

1.0201 

.0000 

.2013 

.0000 

1.0453 

.0000 

.  3045 

.0000 

0 

1.0150 

0201 

2003 

1.018 

1.0401 

0304 

3o;,o 

1044 

.1 

0.9997 

0100 

1973 

2027 

1.0245 

0605 

298.5 

2077 

o 

0.974.5 

0595 

1923 

3014 

9987 

0900 

2909 

3089 

!3 

.9:J9.-) 

.0784 

.  1854 

.3972 

.9628 

.1186 

.280.5 

.4071 

.4 

8953 

0965 

1767 

4890 

9174 

1460 

2672 

5012 

.5 

8-119 

1137 

1662 

5760 

8627 

1719 

2513 

5903 

.6 

7802 

1297 

1540 

6571 

7995 

1962 

2329 

6734 

.7 

.7107 

.1444 

.1403 

.7318 

.7283 

.2184 

2122 

.7498 

.8 

6341 

1577 

1252 

7990 

6498 

2385 

189§ 

8188 

.9 

5511 

1694 

1088 

8.584 

5648 

2.562 

1645 

8796 

1.0 

4627 

1795 

0913 

9091 

4742 

2714 

1381 

9316 

1.1 

.3696 

.1877 

.0730 

0.9.507 

.3788 

.2838 

.1103 

0.9743 

1.2 

2729 

1940 

0539 

0.9^29 

2796 

2934 

081.5 

1.0072 

1.3 

1734 

1984 

0342 

1.0052 

1777 

3001 

0518 

1.0301 

1.4 

0722 

2008 

0142 

1  0175 

0739 

3038 

0215 

1.0427 

1.5 

0000 

2013 

0000 

1.0201 

0000 

3045 

0000 

1.04.53 

'.n 

X   = 

.6 

X  = 

•  7 

a 

i 

c 

d 

a 

b 

c 

d 

y 

1.185f) 

.0000 

.  6367 

.0000 

1.2552 

.0000 

.75S6 

.0000 

0 

1.1795 

0636 

6335 

1183 

1.2489 

0757 

7518 

12.53 

.1 

1.1618 

1265 

6240 

2355 

1.2301 

1507 

743.5 

2494 

.2 

1 . 1325 

1881 

6082 

3503 

1.1991 

2242 

7247 

3709 

.3 

1.0918 

.2479 

.5864 

.4617 

1.1561 

.29.54 

.6987 

.4888 

.4 

1.0403 

3052 

5587 

5684 

1.1015 

3637 

6657 

6018 

.5 

0.9784 

35!  5 

52.15 

6694 

1.0:'..59 

4283 

6261 

7087 

.6 

0  9067 

4101 

4869 

7637 

0.9600 

4887 

5802 

8086 

.7 

.8259 

.4567 

.4436 

0.8.504 

.874.5 

.5442 

.5285 

0.9004 

.8 

7369 

4987 

3957 

0.9286 

7802 

5942 

4715 

0.9832 

.9 

6405 

5357 

344(1 

0.9975 

6782 

6383 

4099 

1.0.562 

1.0 

5377 

5674 

2888 

1.0.56.5 

5693 

6760 

3441 

1.1186 

1.1 

.4296 

5934 

.2307 

1.1049 

.4.548 

.7070 

.2749 

1.1699 

1.2 

3171 

613.5 

1703 

1.1422 

33.58 

7309 

3029 

1  2094 

1.3 

2015 

6274 

10S2 

1  1682 

2133 

7475 

1289 

1 . 2369 

14 

0839 

6:;5i 

04.50 

1.1825 

0888 

7567 

0537 

1.2.520 

1.5 

0000 

6367 

0000 

1.1855 

0000 

7586 

0000 

1.2552 

\n 

G8 


HYPERBOLIC    FUNCTIONS. 
Table  II.     Values  of  cosh  (x  +  iy)  and  sinh  (jt  +  iy). 


X   = 

.8 

X   = 

•9 

y 

a 

b 

c 

d 

a 

b 

c 

d 

0 

1.3374 

0000 

.8881 

.0000 

1.4:131 

.0000 

1.0265 

.0000 

.1 

1 . 3:508 

0887 

8837 

1335 

1.4259 

1 025 

1.0214 

1431 

.2 

1.3108 

1764 

8:u4 

26:)  7 

1.4045 

2031» 

1.0061 

2847 

.3 

l.'iTTG 

2625 

8481 

3952 

1.3691 

3034 

0.9807 

4235 

.4 

l.?319 

.3458 

.8180 

.5208 

1.3200 

.3997 

.9455 

.5581 

.5 

1.1737 

4258 

7794 

6412 

1  2577 

4921 

9008 

6b7i 

.6 

1 . 1038 

5015 

73o0 

7552 

1  18-28 

5796 

8472 

8092 

.7 

1.0229 

5721 

6793 

8616 

1.0961 

6613 

7851 

9232 

.8 

.9:118 

.6371 

.6188 

0.9595 

.9984 

.7364 

.7152 

1.0280 

.9 

8314 

6957 

5521 

1.0476 

8908 

8041 

6381 

1.1226 

1.0 

7226 

7472 

4798 

1.1254 

7743 

8638 

5546 

1.2059 

1.1 

6067 

7915 

40-28 

1.1919 

6:)00 

9148 

4656 

1.2772 

1.2 

.4S46 

.8-278 

.3218 

1.2465 

.5193 

0.9568 

.3720 

1.3357 

1.3 

3r)78 

8557 

2:176 

1.2887 

3834 

0.9891 

2746 

1  3809 

1.4 

2273 

8752 

1510 

1  3180 

2436 

1.0124 

1745 

1.4122 

1.5 

0946 

8859 

0628 

1.3341 

1014 

1.0239 

0726 

1.4295 

in 

0000 

.8881 

0000 

1.3374 

0000 

1.0265 

0000 

1.4331 

X    = 

1.2 

X   = 

I  3 

y 

a 

b 

C 

(/ 

a 

b 

c 

d 

0 

1.8107 

.0000 

1.5095 

.0000 

1.9709 

.0000 

1.6984 

.0000 

.1 

1.8(tl6 

1507 

1.5019 

18(iS 

1.9611 

1696 

1  6899 

1968 

.2 

1.7746 

29i)9 

1.4794 

35!  IS 

1.9:116 

3374 

1.6645 

3916 

.3 

1.7298 

4461 

1.4420 

5351 

1.88-29 

5019 

1.6225 

5824 

.4 

1.6677 

.587S 

1.3903 

0.70.-,l 

1.8153 

.6614 

1.5643 

0.7675 

.5 

1.5890 

7237 

1.3-247 

0  8681 

1.7296 

8142 

1  4905 

0  9449 

.6 

1.4944 

8523 

1.2458 

1.0-224 

1  6267 

9590 

1  4017 

1.1131 

.7 

1.3849 

9724 

1.1 54f) 

1.1 66f) 

1  5074 

1.0941 

1.2990 

1.2697 

.8 

1.2615 

1.08-28 

1.0.-)17 

1.2989 

1.3731 

1.2183 

1.1833 

1.4139 

.9 

1.12")-, 

1.18-24 

0  9:58:^ 

1.4IS:{ 

1  22.-.1 

1.3304 

1.0557 

1.5439 

1.0 

0.978:? 

1.2702 

0.8156 

1.5-236 

i.oc.iij 

1  4291 

0.9176 

1.6585 

1.1 

0.8213 

1.3452 

0.6847 

1.6137 

0.8940 

1.5i:J6 

0.7704 

1.7565 

1.2 

.6-)61 

1.4069 

.547(1 

1  6S76 

.7142 

1.58:!0 

.6154 

1  8370 

1.3 

4844 

1.454  i 

4088 

1.7447 

5272 

1  6365 

4.-)43 

1  8991 

1.4 

3078 

1.4S75 

2566 

1.7S43 

33.10 

1  6737 

28S7 

1.9422 

1.5 

1281 

1.5057 

1068 

1.80(il 

13!»4 

1.6941 

1201 

1.9660 

\Tt 

0000 

1.5095 

0000 

1.8107 

0000 

1.6984 

0000 

1.9709 

TABLES. 

Table  II.     Values  of  cosh  (x  -|-  iv)  and  sink  (jt  -)-  iy.) 


69 


X   = 

I.O 

X   = 

I.I 

a 

i> 

c 

J 

a 

d 

c 

d 

y 

1.5431 

.0000 

1.1752 

.0000 

1.6685 

.0000 

1.3356 

.0000 

0 

1.5354 

1173 

1.1693 

1541 

1.6602 

1333 

1  3--'90 

1666 

.1 

1.5128 

2335 

1.1518 

3066 

1.6353 

2654 

1.3090 

3315 

.2 

1.4742 

3473 

1.1227 

4560 

1.5940 

3946 

1.2760 

4931 

.3 

1.4213 

.4576 

1.0824 

.6009 

1.5368 

.5201 

1  2302 

0  6498 

.4 

1.3542 

5634 

1.0314 

7398 

1.4643 

6403 

1.1721 

0.7099 

.5 

1.2736 

6636 

0.9699 

8718 

1.3771 

7542 

1  1024 

0.9421 

.6 

1 . 1803 

7571 

0.8988 

9941 

1.2762 

8604 

1.0216 

1.0749 

.7 

1.0751 

0.8430 

.8188 

1  1069 

1.1625 

0  9581 

.9306 

1  1969 

.8 

0  9592 

0  9206 

7305 

1.2087 

1.0372 

1.0462 

8302 

1.3(»70 

.9 

0.8337 

0.9,s89 

6350 

1.298.5 

0.9015 

1.1239 

7217 

1.4040 

1.0 

0.6999 

1.0473 

5331 

1.3752 

0  7568 

1.1903 

6058 

1.4870 

1.1 

.5592 

1.0953 

.4258 

1.4382 

.6046 

1  2449 

.4840 

1.5551 

1.2 

4128 

1.1324 

3144 

1.4869 

4463 

1.2870 

3)73 

1.6077 

1.3 

2623 

1.1581 

1998 

1.5213 

2836 

1.3162 

2270 

1.6442 

1.4 

1092 

1.1723 

0831 

1  5392 

1180 

1.3323 

0945 

1.6643 

1.5 

0000 

1.1752 

0000 

1  5431 

0000 

1.3356 

0000 

1.6685 

\n 

X   = 

1-4 

X  = 

1.5- 

a 

d 

<r 

r/ 

a 

b 

c 

d 

y 

2.1.509 

.0000 

1.9043 

.0000 

2.3524 

.0000 

2.1293 

.0000 

0 

2.1401 

1901 

1.8948 

2147 

2  3413 

2126 

2.1187 

2348 

.1 

2.1080 

3783 

1.8663 

4273 

2  3055 

4280 

2.0868 

4674 

.2 

2.0548 

5628 

1.8192 

6356 

2  2473 

6292 

2.0342 

6951 

.3 

1  9811 

0.7416 

1  7540 

0.8376 

2.1667 

0.8202 

1.9612 

0.9161 

.4 

1.8876 

0.9130 

1  6712 

1.0312 

2.0644 

1.0208 

1.8686 

1.1278 

.5 

1 . 7752 

1.0753 

1  5713 

1.2145 

1.9415 

1  2023 

1 .7574 

1.3283 

.6 

1.6451 

1  2268 

1.4565 

1.3856 

1.7992 

1.3717 

1.6286 

1.5155 

.7 

1.4985 

1.3661 

1.3268 

1.5430 

1.6389 

1.5275 

1.4835 

1.6875 

.8 

1.3370 

1  4917 

1.1838 

1.6849 

1.4623 

1.6679 

1.3236 

1.8427 

.9 

1.1622 

1.6024 

1.0289 

1.8099 

1.2710 

1  7917 

1.150.5 

1.9795 

1.0 

0.9756 

1.6971 

0.8638 

1.9168 

1.0671 

1.8976 

0  9659 

2.0965 

1.1 

.7794 

1.7749 

.6900 

2  0047 

.8524 

1  0846 

.7716 

2.1925 

1.2 

5754 

1.8349 

5094 

2.0725 

6293 

2.0517 

5606 

2.2667 

1.3 

3656 

1.8766 

3237 

2.1196 

3998 

2.0983 

3619 

2  3182 

1.4 

1522 

1.8996 

1347 

2.1455 

1664 

2.1239 

1506 

2  3465 

1.5 

.0000 

1.9043 

0000 

2.1509 

.0000 

2.1293 

.0000 

2.3524 

\Tt 

70 


HYPERBOLIC    FUNXl  IONS. 


Table  III. 


u 

gd« 

0^ 

u 

!   gd« 

r 

ti 

gd  « 

6° 

00 

.0000 

0.000 

.60 

.5069 

32.483 

1.50 

1.1317 

o 

64.843 

.02 

0300 

1.146 

.63 

5837 

33.444 

1.55 

1.1.525 

66.034 

.04 

0400 

2.291 

.64 

6003 

34.395 

1  60 

1.1724 

67.171 

.06 

0600 

3  436 

.66 

6167 

35  336 

1  65 

1.1913 

68.257 

.08 

0799 

4.579 

.68 

6329 

36.265 

1.70 

1.2094 

69.294 

.10 

.0998 

5.720 

.70 

.6489 

37.183 

1.75 

1.2267 

70.284 

.13 

1197 

6.859 

.73 

6648 

38.091 

1  80 

1.2432 

71  228 

.14 

1395 

7.995 

.74 

6804 

38.987 

1.85 

1.2589 

72.128 

.16 

1593 

9.128 

.76 

6958 

39.872 

1.90 

1.2739 

72.987 

.18 

1790 

10.25» 

.78 

7111 

40.746 

1.95 

1.2881 

73.805 

.30 

.1987 

11  384 

.80 

.7261 

41  608 

2.00 

1.3017 

74.584 

.?.3 

3183 

12.505 

.83 

7410 

42460 

2.10 

1.3271 

76.037 

.34 

2377 

13  621 

.84 

7557 

43.299 

2.20 

1.3501 

77.354 

.26 

2571 

14.732 

.86 

7702 

44  128 

2  30 

1.3710 

78.519 

.28 

2764 

15.837 

.88 

7844 

44  944 

3.40 

1.3899 

79.633 

.30 

.2956 

16.937 

.90 

.7985 

45  750 

2  50 

1.4070 

80  615 

.'61 

3147 

18.030 

.92 

8123 

46.544 

2  60 

1.4227 

81.513 

.34 

3336 

1.9  116 

.94 

8260 

47.326 

2.70 

1.4366 

82.310 

.36 

3.V25 

20.195 

.96 

8394 

48.097 

2  80 

1.4493 

83.040 

.38 

3712 

.3897 

21.267 
22.331 

.98 

8528 
.8658 

48.857 
49.605 

2.90 
3  00 

1.4609 
1.4713 

83.707 

.40 

1  00 

84.301 

.43 

4082 

23.386 

1.05 

8976 

51.428 

3.10 

1.4808 

84.841 

.44 

4264 

24.434 

r.io 

9J81 

53  178 

3.20 

1.4894 

85.336 

.46 

4146 

25.473 

1  15 

9575 

54  860 

3.30 

1.4971 

85.775 

.48 

4626 

26  503 

1.20 

9857 

56  476 

3.40 

1.5041 

86.177 

.50 

.4804 

27  524 

1.25 

1.0127 

58.026 

3.50 

1.5104 

86.541 

.53 

4980 

28  5:!5 

1.30 

1.0387 

59.511 

3.60 

1  5162 

86.870 

.54 

5155 

29  5i7 

1.35 

1.0635 

60.933 

3  70 

1.5214 

87  168 

.56 

5328 

30.529 

1.40 

1.0873 

62.295 

3.80 

1.5261 

87.437 

.58  1 

55(J0 

31.511 

1.45 

1. 1100 

63.598 

3.90 

1.5303 

87.681 

Table  IV. 

u 

gd  u 

log  sinh  ti 

log  cosh  u 

~5.5 

gd  u 
1.5626 

log  sinh  M 

log  cosh  u 

4.0 

1.5343 

1  43(i0 

1.4363 

2.08758 

2.08760 

4.1 

1.5377 

1.4795 

1.4797 

5.6 

1.5634 

2.13101 

2.13103 

4.3 

1.5408 

1..5229 

1.5231 

5.7 

1.5(341 

2.17444 

2.17445 

4.3 

1  5437 

1..5664 

1.5665 

5.8 

1.5648 

2.21787 

2  21788 

4.4 

1.5463 
1..54S6 

1  6098 
].6.5:',2 

1.6099 
1.6.-)33 

5.9 

1.5653 
1  56.58 

2.26130 
2.30473 

2.26131 

4.5 

60 

2.30474 

4  6 

1.5507 

1  6967 

1.6968 

62 

1.5667 

2.391.59 

2.39160 

4.7 

1  5.526 

1.7401 

1.7402 

6.4 

1 . 5675 

2.47S45 

2.47846 

4.8 

1.5543 

1.7836 

1.7836 

6.6 

1.5681 

2.56531 

2.56531 

4.9 

1.55-)9 
1  5573 

1.8270 
1.S704 

1.8270 

1.870.5 

6  8 

1.5686 
1.5690 

2.6.5217 
2.73903 

2.65217 

5.0 

7.0 

2.73903 

5.1 

1  558(5 

1.9139 

1  9139 

7.5 

1.5697 

2.9.5618 

3.95618 

5.2 

1.. 5.598 

1.9573 

1.9.-)73 

8.0 

1.5701 

3.17333 

3.17333 

5.3 

1  5608 

2.0007 

2  (1007 

8.5 

1.5704 

3.39047 

3.39047 

5.4 

1  5618 

2.0443 

2.0442 

9.0 

1.5705 

3.60762 

3.60762 

00 

1.5708 

00 

00 

APPENDIX. 


A.     HISTORICAL   AND    BIBLIOGRAPHICAL. 

What  is  probably  the  earUest  suggestion  of  the  analogy  between 
the  sector  of  the  circle  and  that  of  the  hyperbola  is  found  in  Newton's 
Principia  (Bk.  2,  prop.  8  et  seq.)  in  connection  with  the  solution  of  a 
dynamical  problem.  On  the  analytical  side,  the  first  hint  of  the  modi- 
fied sine  and  cosine  is  seen  in  Roger  Cotes'  Harmonica  Mensurarum 
(1722),  where  he  suggests  the  possibility  of  modifying  the  expression 
for  the  area  of  the  prolate  spheroid  so  as  to  give  that  of  the  oblate  one, 
by  a  certain  use  of  the  operator  V—i,  The  actual  inventor  of  the 
hyperbolic  trigonometry  was  Vincenzo  Riccati,  S.J.  (Opuscula  ad  res 
Phys.  et  Math,  pertinens,  Bononia^,  1757).  He  adopted  the  notation 
Sh.^,  Ch.(y6  for  the  hyperbolic  functions,  and  Sc.^,  Cc.0  for  the  cir- 
cular ones.  He  proved  the  addition  theorem  geometrically  and  derived 
a  construction  for  the  solution  of  a  cubic  equation.  Soon  after,  Daviet 
de  Foncenex  showed  how  to  interchange  circular  and  h}-perbolic  func- 
tions by  the  use  ofv  —  i,  and  gave  the  analogue  of  De  Moivre's  theorem, 
the  work  resting  more  on  analogy,  however,  than  on  clear  definition 
(Reflex,  sur  les  quant,  imag.,  Miscel.  Turin  Soc,  Tom.  i).  Johann 
Heinrich  Lambert  systematized  the  subject,  and  gave  the  serial  devel- 
opments and  the  exponential  expressions.  He  adopted  the  notation 
sinh  u,  etc.,  and  introduced  the  transcendent  angle,  now  called  the 
gudermanian,  using  it  in  computation  and  in  the  construction  of  tables 
(1.  c.  page  30).  The  important  place  occupied  by  Gudermann  in  the 
history  of  the  subject  is  indicated  on  page  30. 

The  analogy  of  the  circular  and  hyperbolic  trigonometry  naturally 
played  a  considerable  part  in  the  controversy  regarding  the  doctrine 
of  imaginaries,  which  occupied  so  much  attention  in  the  eighteenth  cen- 
tury, and  which  gave  birth  to  the  modern  theory  of  functions  of  the 


72  HYPERBOLIC    FUNCTIONS. 

complex  variable.  In  the  growth  of  the  general  complex  theory,  the 
importance  of  the  "  singly  periodic  functions"  became  still  clearer,  and 
was  gradually  developed  by  such  writers  as  Ferroni  (Magnit.  expon. 
log.  et  trig.,  Florence,  1782);  Dirksen  (Organon  der  tran.  Anal.,  Ber- 
lin, 1845);  Schellbach  (Die  einfach.  period,  funkt.,  Crelle,  1854);  Ohm 
(Versuch  eines  volk.  conseq.  Syst.  der  Math.,  Nlirnberg,  1855);  Hoiiel 
(Theor.  des  quant,  complex,  Paris,  1870).  Many  other  writers  have 
helped  in  systematizing  and  tabulating  these  functions,  and  in  adapting 
them  to  a  variety  of  applications.  The  following  works  may  be  espe- 
cially mentioned:  Gronau  (Tafeln,  1862,  Theor.  und  Anwend.,  1865); 
Forti  (Tavoli  e  teoria,  1870);  Laisant  (Essai,  1874);  Gunther  (Die 
Lehre  .  .  .  ,  1S81).  The  last-named  work  contains  a  very  full  history 
and  bibliography  with  numerous  applications.  Professor  A.  G.  Green- 
hill,  in  various  places  in  his  writings,  has  shown  the  importance  of  both 
the  direct  and  inverse  hyperbolic  functions,  and  has  done  much  to  pop- 
ularize their  use  (see  Diff.  and  Int.  Calc,  1891).  The  following  articles 
on  fundamental  conceptions  should  be  noticed:  Alacfarlane,  On  the 
definitions  of  the  trigonometric  functions  (Papers  on  Space  Analysis, 
N.  Y.,  1894);  Haskell,  On  the  introduction  of  the  notion  of  hyperbolic 
functions  (Bull.  N.  Y.  M.  Soc,  1895).  Attention  has  been  called  in 
Arts.  30  and  37  to  the  work  of  Arthur  E.  Kennelly  in  applying  the 
hyperbolic  complex  theory  to  the  plane  vectors  which  present  them- 
selves in  the  theory  of  alternating  currents;  and  his  chart  has  been 
described  on  page  44  as  a  useful  substitute  for  a  numerical  complex 
table  (Proc.  A.  I.  E.  E.,  1895).  It  may  be  worth  mentioning  in  this 
connection  that  the  present  writer's  complex  table  in  Art.  39  is  believed 
to  be  the  earliest  of  its  kind  for  any  function  of  the  general  argum.ent 
X  +  iy-     (See  Appendix,  C.) 

B.     EXPONENTIAL   EXPRESSIONS   AS   DEFINITIONS. 

For  those  who  wish  to  start  with  the  exponential  expressions  as  the 
defmitions  of  sinh  ii  and  cosh  »,  as  indicated  on  page  25,  it  is  here  pro- 
posed to  show  how  these  defmitions  can  be  easily  brought  into  direct 
geometrical  relation  with  the  hyperbolic  .sector  in  the  form  .v/<7  =  cosh 
S/K,  y/b  =  sm\\  S/K,  by  making  use  of  the  identity  cosh-  z<  —  sinh^  "=  i, 
and  the  differential  relations  d  cosh  H  =  sinh  u  du,  d  sinh  H  =  cosh  u  dii, 
which  arc  themselves  immediate  con.sequcnces  of  tho.se  exponential 
definitions.     Let  0.1,  the   initial  radius  of  the   hyperbolic   sector,  be 


EXPONENTIAL    EXPRESSIONS   AS    DEFINITIONS.  73 

taken  as  axis  of  .r,  and  its  conjugate  radius  OB  as  axis  of  y;  let  OA  =a, 
OB  =  b,  angle  AOB  =  aj,  and  area  of  triangle  AOB  =  K,  then  K= 
^ab  sin  co.  Let  the  coordinates  of  a  point  P  on  the  hyperbola  be  x 
and  y,  then  x'^/a"  —  y~/b''^==i.  Comparison  of  this  equation  with  the 
identity  cosh^  7/  — sinh^  ii=i  permits  the  two  assumptions  .T/a  =  cosh  m 
and  y/^'  =  sinh  u,  wherein  u  is  a  single  auxiliary  variable;  and  it  now 
remains  to  give  a  geometrical  interpretation  to  ii,  and  to  prove  that 
u=S/K,  wherein  5  is  the  area  of  the  sector  OAP.  Let  the  coordinates 
of  a  second  point  Q  be  .v+ J.v  and  y+Jy,  then  the  area  of  the  triangle 
POQ  is,  by  analytic  geometry,  ^(.vJj— yJ.v)sin  <6».  Now  the  sector 
POQ  bears  to  the  triangle  POQ  a  ratio  whose  limit  is  unity,  hence  the 
differential  of  the  sector  S  may  be  written  dS=^{x  dy—y  dx)s'in  oj= 
i^ab  s'm  io{cos\i"  u  —  s'mh~  ii)du  =  K  du.  By  integration  S=Kh,  hence 
u  =  S/K,  the  sectorial  measure  (p.  lo);  this  establishes  the  fundamental 
geometrical  relations  .Y/a  =  cosh  S/K,  j/6  =  sinh  S/K. 

C.     RECENT  TABLES  AND   APPLICATIONS. 

The  most  extensive  tables  of  hyperbolic  functions  of  real 
arguments  are  those  published  by  the  Smithsonian  Institution,  pre- 
pared by  G.  F.  Becker  and  C.  E.  Van  Orstrand  (igog). 

For  complex  arguments  the  most  elaborate  tables  are  those  of 
Professor  A.  E.  Kennelly:  "Tables  of  Complex  Hyperbolic  and 
Circular  Functions  "  (Harvard  University  Press,  igi4). 

Three-digit  tables  of  sinh  and  cosh  of  x-^iy,  up  to  x=i  and 
y=i  by  steps  of  .01,  are  given  by  W.  E.  Miller  in  a  paper, 
"  Formulae,  Constants,  and  Hyperbolic  Functions  for  Transmission- 
line  Problems  "  in  the  General  Electric  Review  Supplement,  Schen- 
ectady, N.  Y.,  May,  igio. 

There  are  interesting  applications  and  an  extensive  bibliography 
in  Professor  Kennelly 's  treatise  on  "  The  Application  of  Hyperbolic 
Functions  to  Electrical  Engineering  Problems "  (University  of 
London  Press,  igi2). 

It  should  be  noted  that  this  author  uses  the  term  "  hyperbolic 
angle  "  for  "  hyperboHc  sectorial  measure,"  the  analogy  being  due 
to  the  fact  that  the  "  sectorial  measure  "  for  the  circle  and  ellipse 
is  an  actual  angle  (p.  11).  The  convenient  term  "  angloid  "  has 
been  suggested  by  Professor  S.  Epsteen. 


INDEX. 


Addition-theorems,  pages  i6,  40. 
Admittance  of  dielectric,  56. 
Algebraic  identity,  41. 
Alternating  currents,  38,  46,  55. 
Ambiguity  of  value,  13,  16,  45. 
Amplitude,  hyperbolic,  31. 

of  complex  number,  46. 
Anti-gudermanian,  28,  30,  47,  31,  52. 
Anti-hyperbolic  functions,  16,  22,  25,  29, 

35.  45- 
Applications,  46  et  seq. 
Arch,  48,  51. 

Areas,  8,  9,  14,  36,  37,  60. 
Argand  diagram,  43,  58. 

Bassett's  Hydrodynamics,  61. 
Beams,  flexure  of,  54. 
Becker  and  Van  Orstrand,  73. 
Bedell  and  Crehore,  38,  56. 
Byerly's  Fourier  Series,  etc.,  61,  63. 

Callet's  Tables,  63. 
Capacity  of  conductor,  55. 
Catenary,  47. 

of  uniform  strength,  49. 
Elastic,  48. 
Cayley's  Elliptic  Functions,   30,   31. 
Center  of  gravity,  61. 
Characteristic  ratios,  10. 
Chart  of  hyperbolic  functions,  44,  58. 

Mercator's,  53. 
Circular  functions,  7,  11,  14,  18,  21,  24, 

29,  35.  41.  43- 
of  complex  numbers,  39,  41,  42. 
of  gudermanian,  28 


Complementary  triangles,  10. 
Complex  numbers,  38-46. 

Applications  of,  55-60. 

Tables,  62,  66. 
Conductor   resistance   and    impedance, 

58- 
Construction  for  gudermanian,  30. 
of  charts,  43. 
of  graphs,  32. 
Convergence,  23,  25. 
Conversion-formulas,  18. 
Corresponding  points  on  conies,  7,  28. 

sectors  and  triangles,  9,  28. 
Currents,  alternating,  55. 
Curvature,  50,  52,  60. 
Cotes,  reference  to,  71. 

Deflection  of  beams,  54. 

Derived  functions,  20,  22,  30. 

Difference  formula,  16. 

Differential  equation,  21,  25,  47,  49,  51, 

52,  57- 
Dirksen's  Organon,  71. 
Distributed  load,  55. 

Electromotive  force,  55,  58. 
Elimination  of  constants,  21. 
Ellipses,  chart  of  confocal,  43. 
Elliptic  functions,  7,  30,  31. 

integrals,  7,  31. 

sectors,  7,  31. 
Equations,  Differential  (see). 

Numerical,  35,  48,  50. 
E volute  of  tractory,  52. 
Expansion  in  series,  23,  25,  31. 


76 


INDEX. 


Exponential  expressions,  24,  25,  72. 

Ferroni,  reference  to,  71. 
Flexure  of  beams,  53. 
Foncenex,  reference  to,  71. 
Forti's  Tavoli  e  teoria,  63,  71. 
Fourier  series,  55,  61. 
Function,  anti-gudermanian  (see). 

anti-hyperbolic  (see). 

circular  (sec). 

elliptic  (see). 

gudermanian  (see). 

hyperbolic,  defined,  11. 

of  complex  numbers,  38. 

of  pure  imaginarics,  41. 

of  sum  and  difference,  16. 

periodic,  44. 

Geipel  and  Kilgour's  Electrical  Hand- 
book, 63. 
Generalization,  41. 
Geometrical  interpretation,  37. 

treatment     of     hyperbolic 
functions,  yetseq.,  16. 
Glaishcr's  exponential  tables,  63. 
Graphs,  32. 
Greenhill's  Calculus,  72. 

Elliptic  Functions,  7. 
Gronau's  Tafeln,  63,  72. 

Theor.  und  Anwend.,  72. 
Gudermann's  notation,  30. 
Gudermanian,  angle,  29. 

function,  28,  31,  34,  47>  53,  63,  70. 
Gunther's  Die  Lehre,  etc.,  61,  71. 

Haskell  on  fundamental  notions,  72. 
Houel's  notation,  etc.,  30,  31,  71. 
Hyperbola,  7  et  seq.,  30,  37,  44,  60. 
Hyperbolic  functions,  defined,  11. 

addition-theorems  for,  16. 

applications  of,  46  et  seq. 

derivatives  of,  20. 

expansions  of,  23. 

exponential  expressions  for,  24. 

graphs  of,  32. 

integrals  involving,  35. 


Hyperbolic  functions  of  complex  num- 
bers, 38  el  seq. 
relations  among,  12. 
relations  to  gudermanian,  29. 
relations  to  circular  functions,  29, 42. 
tables  of,  64  et  seq. 
variation  of,  20. 

Imaginary,  see  complex. 

Impedance,  34. 

Integrals,  35. 

Interchange  of  hyperbolic  and  circular 

functions,  42. 
Interpolation,  30,  48,  50,  59,  62. 
Intrinsic  equation,  38,  47,  49,  51. 
Involute  of  catenary,  48. 
of  tractory,  50. 

Jones'  Trigonomet.y,  52. 

Kennclly  on  alternating  currents,  38,  58. 
Kcnnelly's  chart,  46,  58;  treatise,  73. 

Liisant's  Essai,  etc.,  61,  71. 
Lambert's  notation,  30. 

place  in  the  history,  70. 
Leakage  of  conductor,  55. 
Limiting  ratios,  19,  23,  32. 
Logarithmic  curve,  60. 

expressions,  27,  32. 
Love's  elasticity,  61. 
Loxodrome,  52. 

Macfarlane  on  definitions,  72. 

Maxwell's  Electricity,  61. 

Measure,  defined,  8;  of  sector,  9  et  seq. 

IMercator's  chart,  53. 

Miller,  W.  E.,  Tables,  etc.,  73 

Modulus,  31,  46. 

Moment  of  inertia,  61. 

Multiple  values,  13,  16,  45. 

Newton,  reference  to,  71. 
Numbers,  complex,  38  et  seq. 

Ohm,  reference  to,  71. 
Operators,  generalized,  39,  56. 

Parabola,  38,  61. 
Periodicity,  44,  62. 


INDEX. 


Ti 


Permanence  of  equivalence,  41. 

Phase  angle,  56,  59. 

Physical  problems,  21,  38,  47  et  seq. 

Potential  theory,  61. 

Product -series,  43. 

Pure  imaginary,  41. 

Ratios,  characteristic,  10. 

limiting,  19. 
Rayleigh's  Theory  of  Sound,  61. 
Reactance  of  conductor,  58. 
Reduction  formula,  37,  38. 
Relations  among  functions,  12,  29,  42. 
Resistance  of  conductor,  56. 
Rhumb  line,  53. 
Riccati's  place  in  the  history,  71. 

Schellbach,  reference  to,  71. 
Sectors  of  conies,  9,  28. 


Self-induction  of  conductor,  55. 

Series,  23,  31. 

Spheroid,  area  of  oblate,  58 

Spiral  of  Archimedes,  60. 

Steinmetz  on  alternating  currents,  38. 

Susceptancc  of  dielectric,  58. 

Tables,  62,  73. 

Terminal  conditions,  54,  58,  60. 

Tractory,  48,  51. 

Van  Orstrand,  C.  E.,  Tables,  73. 
Variation  of  hyperbolic  functions,  lA. 
Vassal's  Tables,  63. 
Vectors,  38,  56. 
Vibrations  of  bars,  61, 

Wheeler's  Trigonometry,  65. 


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